inequalities problem solving

Solving Inequalities

Sometimes we need to solve Inequalities like these:

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

x > 10

How to Solve

Solving inequalities is very like solving equations , we do most of the same things ...

... but we must also pay attention to the direction of the inequality .

Some things can change the direction !

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

Safe Things To Do

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

But these things do change the direction of the inequality ("<" becomes ">" for example):

• Multiply (or divide) both sides by a negative number
• Swapping left and right hand sides

Example: 2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality :

12 > 2y+7

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra ), like this:

Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 − 3 < 7 − 3    

And that is our solution: x < 4

In other words, x can be any value less than 4.

What did we do?

And that works well for adding and subtracting , because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5

If we subtract 5 from both sides, we get:

12 − 5 < x + 5 − 5    

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying ).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number :

Example: 3y < 15

If we divide both sides by 3 we get:

3y /3 < 15 /3

And that is our solution: y < 5

Negative Values

Well, just look at the number line!

For example, from 3 to 7 is an increase , but from −3 to −7 is a decrease.

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality !

−2y < −8

−2y /−2 > −8 /−2

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Example: bx < 3b

It seems easy just to divide both sides by b , which gives us:

... but wait ... if b is negative we need to reverse the inequality like this:

But we don't know if b is positive or negative, so we can't answer this one !

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b :

• if b is 1 , then the answer is x < 3
• but if b is −1 , then we are solving −x < −3 , and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b .

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

Example: x−3 2 < −5.

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 2 ×2 < −5  ×2  

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −10 + 3    

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Example: −2 < 6−2x 3 < 4

First, let us clear out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

−6 < 6−2x < 12

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction .

6 > x > −3

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

−3 < x < 6

• Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
• Multiplying or dividing both sides by a negative number
• Don't multiply or divide by a variable (unless you know it is always positive or always negative)

How to Solve Inequalities—Step-by-Step Examples and Tutorial

September 13, 2023 by Anthony Persico

How to Solve Inequalities Explained

Step-by-step guide: how to solve an inequality equation.

Free Step-by-Step Guide: How to solve inequalities and inequality equations

In math, an inequality is a symbol that is used to represent the relationship between two values or expressions that are not necessarily equal to each other. There are four types of inequality symbols:

> : greater than

< : less than

≥ : greater than or equal to

≤ : less than or equal to

Understanding how to solve inequalities is an important math skill that all students will need to be successful in algebra and beyond. To acquire this skill, you will have to build upon your knowledge of solving equations and extend that understanding to solving inequalities.

This free Step-by-Step Guide on How to Solve Inequalities covers the following:

What is an inequality?

How to Solve Inequalities Example #1 (basic)

How to Solve Inequalities Example #2 (basic)

When do inequality signs reverse direction?

How to Solve Inequalities Example #3 (complex)

How to Solve Inequalities Example #4 (complex)

How to Solve Inequalities with Fractions: Example #5

How to Solve Inequalities with Fractions: Example #6

How to Solve Inequalities Extra Practice: Example #7

How to Solve Inequalities Extra Practice: Example #8

Free Solving Inequalities Worksheet (w/ Answers)

While learning how to solve an inequality is a little trickier than learning how to solve an equation, you can easily learn how to solve inequalities by working through this guide. By working through the step-by-step practice problems below, you will gain plenty of helpful practice with solving inequalities, which will put you on the path of being able to solve any inequality with ease.

While you are probably eager to dive into a few practice problems, let’s start off with a quick review of a few key vocabulary terms that you should deeply understand before you begin learning how to solve an inequality.

Figure 01: There are four types of inequality symbols: > : greater than, < : less than, ≥ : greater than or equal to, and ≤ : less than or equal to.

As said earlier, in math, an inequality is a relationship between two values or expressions that are not equal to each other. Because the values or expressions are not equal to each other, we have to use an inequality sign instead of an equals (=) sign.

You already know that there are four types of inequalities: > : greater than, < : less than, ≥ : greater than or equal to, and ≤ : less than or equal to (these four types of inequalities are illustrated in Figure 01 above).

Basic inequalities do not need to be solved since the variable is already by itself.

For example, consider the inequalities x>7 and y ≤-3 .

These inequalities can be thought of as solved because the variables are already on their own. These “solved” inequalities are illustrated in Figure 02 below.

Figure 02: How to solve inequalities. What do solved inequalities look like?

But, what if we had an inequality that looked like this: x - 3 > 7

Since the variable, x, is not by itself, this inequality still needs to be solved, and the remainder of this guide will show you exactly how to do that.

Now that you are familiar with the key vocabulary terms, it’s time to look at a few examples.

How to Solve Inequalities Example #1

Example: x - 3 > 7

Just as you would solve an equation, to solve an inequality, you must use inverse operations to isolate the variable, which, in this example, is x.

You can isolate x easily by adding 3 to both sides of the inequality sign as follows:

x - 3 > 7

x -3 + 3 > 7 +3

Now the inequality is solved! The answer is x>10. The step-by-step procedure to solving this first example is illustrated in Figure 03 below.

Figure 03: How to solve an inequality: x-3>7

This first example of how to solve inequalities was relatively simple and only took one step to solve. Now, let’s move onto solving a slightly more difficult inequality.

How to Solve Inequalities Example #2

Example: 3x + 8 < 26

Again, to solve the inequality you have to isolate the variable x by performing inverse operations as follows:

3x + 8 < 26

3x - 8 < 26 - 8

3x/3 < 18/3

Solving this example required two steps (step one: subtract 8 from both sides; step two: divide both sides by 3). The result is the solved inequality x<6.

The step-by-step procedure to solving example #2 is illustrated in Figure 04 below.

Figure 04: How to solve an inequality: 3x+8<26

When Do Inequality Signs Reverse Directions?

The first two examples of solving inequalities are considered basic and they can both be solved in two steps or less.

However, more complex inequalities can include instances when the inequality sign will reverse direction in one of the following ways:

> becomes <

< becomes >

≥ becomes ≤

≤ becomes ≥

So, when do you have to worry about reversing the direction of an inequality sign?

Let’s start by listing all of the instances when you are safe and do not have to change the direction of the inequality sign.

Whenever you add or subtract a number from both sides of the inequality

Whenever you multiply or divide both sides of the inequality by a positive number

Both of these cases occurred in examples #1 and #2, which is why we did not have to change the direction of the inequality sign.

However, you do have to change the direction of the inequality sign under these instances:

🔄 Whenever you swap the position of the left side of the inequality with the right side of the inequality

🔄 Whenever you multiply or divide both sides of the inequality by a negative number

These rules are illustrated in Figure 05 below.

Figure 05: When do inequality signs reverse direction?

Now, let’s take a look at a few examples of solving inequalities where you will have to reverse the direction of the inequality sign.

How to Solve Inequalities Example #3

Example: 17 ≤ 4x + 1

This example is a bit different because the variable, x, is on the right side of the equal sign and we typically express solved inequalities with the variable on the left.

You can still solve this problem the same way that you solved example #2, but first you must swap the left side of the inequality with the right side. However, remember that swapping the position of the left side and the right side requires you to also reverse the direction of the inequality as follows:

17 ≤ 4x + 1 ➙ 4x + 1 ≥ 17

⚠️ Notice that the ≤ reversed position to become ≥ !

Now, you can solve 4x + 1 ≥ 17 as follows:

4x + 1 ≥ 17

4x + 1 -1 ≥ 17 -1

4x/4 ≥ 16/4

The inequality has been solved and the final result is x ≥ 4.

The steps for solving example #3 are detailed in Figure 06 below.

Figure 06: How to solve an inequality when the variable is on the right.

Now that we have covered the first instance of when you have to reverse the direction of an inequality sign, let’s see an example of the second instance—Whenever you multiply or divide both sides of the inequality by a negative number.

How to Solve Inequalities Example #4

Example: -5y < 30

Solving this inequality will require only step to isolate the variable, y, on the left side of the inequality. However, since that one step involves multiplying or dividing by a negative number, you must also reverse the direction of the inequality sign as follows:

-5y < 30

-5y/-5 > 30/-5 (⚠️ since you divided by a negative , < becomes >)

The inequality has been solved and the final result is y > -6.

The steps for solving example #4 are detailed in Figure 07 below.

Figure 07: How to solve an inequality when you have to multiply or divide both sides by a negative number.

How to Solve Inequalities with Fractions

Now that you have some more experience with solving inequalities and you know when you have to reverse the direction of the inequality sign, let’s take a look at two more complex examples that involve fractions.

How to Solve Inequalities with Fractions Example #5

Example: x/3 - 6 < 2

When it comes to solving inequalities with fractions, the same strategy as the previous examples will apply. To solve, you have to use inverse operations to isolate the variable on the left side of the equation.

And, if you have to swap the positions of the left and right sides of the inequality or if you multiply or divide both sides by a negative number, then you will have to reverse the direction of the inequality sign.

You can solve this inequality as follows:

x/3 - 6 < 2

x/3 -6 +6 < 2 + 6

3 x (x/3) < 3 x (8)

The inequality has been solved and the final result is x < 24 .

* Note that this example did not require you to reverse the direction of the inequality sign since you did not have to swap positions or multiply/divide by a negative number.

The step-by-step process for solving example #5 is shown in Figure 08 below.

Figure 08: Solving Inequalities with Fractions Example #5

How to Solve Inequalities with Fractions Example #6

Example: (x-7)/-3 ≥ 4

This next example is the most complex one we have seen so far, but our strategy of using inverse operations to isolate the variable remains the same.

(x-7)/-3 ≥ 4

(x-7)/-3 x -3 ≥ 4 x -3 (multiply both sides by -3 to get rid of the fraction)

x-7 ≤ -12 (⚠️ because you multiplied both sides by a negative, ≥ becomes ≤)

x-7 +6 ≤ -12 +7

The inequality has been solved and the final result is x ≤ -5 .

⚠️ Note that we did have to reverse the inequality sign from ≥ to ≤ because we multiplied both sides by negative 3.

The step-by-step process for solving example #5 is shown in Figure 09 below.

Figure 09: Figure 08: Solving Inequalities with Fractions Example #6

As we enter the last section of this step-by-step guide to solving inequalities, let’s take a look at two more multi-step examples. If you can solve these next two problems using the previously discussed strategies, then you will be able to solve almost any problem related to solving inequalities.

How to Solve Inequalities with Fractions Example #7

Example: -6 > (7y-5)/9

This example will require a few steps in order to isolate the variable x, but we can still used inverse operations to solve as follows:

-6 > (7y-5)/9 ➙ (7y-5)/9 < -6

⚠️ Notice that the variable, y, is on the right side of the inequality sign, so we will start by reversing the positions of the left side and the right side, which also means that we have to reverse the direction of the inequality sign.

(7y-5)/9 < -6

(7y-5)/9 x 9 < -6 x 9

7y-5 < -54

7y -5 +5 < -54 + 5

7y < -49

7y ÷ 7 < -49 ÷ 7

The inequality has been solved and the final result is y < -7 .

Note that we did not have to reverse the inequality sign because the number that we multiplied both sides of the equation by was a positive 9.

The step-by-step process for solving example #7 is shown in Figure 10 below.

Figure 10: Solving Inequalities with Fractions Example #7

How to Solve Inequalities with Fractions Example #8

Are you ready for one final example of how how to solve inequalities with fractions? This one is a little tricky, but you have all of the tools that you need to solve it as long as you take it step-by-step.

Example: -5x - 6 ≤ (x+17)/-2

The first thing that you are probably noticing is that there are x’s on both sides of the inequality sign. Since the goal is to isolate x on the left side of the inequality, you will have to use inverse operations to get x by itself.

Let’s start by getting rid of the fraction on the right side of the inequality by multiplying both sides by -2.

-2(-5x - 6) ≤ (x+17)/-2(-2)

To solve the right side of the inequality, you will have to use the distributive property to multiply both terms (-5x-6) by -2.

10x + 12 ≥ x+17 ⚠️

⚠️ And because we just multiplied both sides by a negative number, we had to reverse the direction of the inequality sign, so ≤ became ≥

Now we can continue solving this inequality by using inverse operations:

10x + 12 ≥ x+17

10x + 12 -12 ≥ x +17 -12 (move the constants to the right side)

10x ≥ x + 5

10x -x ≥ x -x +5 (move the variables to the left side)

9x ÷ 9 ≥ 5 ÷ 9 (isolate x)

The inequality has been solved and the final result is x ≥ 5/9 .

Note that it’s totally fine for our final result to be a fraction.

The step-by-step process for solving example #8 is shown in Figure 11 below.

Figure 10: Solving Inequalities with Fractions Example #8

Are you looking for some extra independent practice on how to solve inequalities and how to solve inequalities with fractions?

You can use the link below to download a free solving inequalities worksheet pdf file that includes a complete answer key. We recommend that you work through each problem on your own keeping this guide close by as a reference.

▶ Download your free Solving Inequalities Worksheet (One-Step) PDF

▶ Download your free Solving Inequalities Worksheet (Two-Steps) PDF

Preview: Solving Inequalities Worksheet PDF

While you work through the practice problems, keep the lesson summary main points below in mind.

Conclusion: How to Solve Inequalities

Like equations, inequalities can be solved by using inverse operations to isolate a variable on the left side of the inequality.

Solving an inequality by adding/subtracting numbers on both sides of the inequality or by multiplying/dividing both sides by a positive number does not result in a reversal of the inequality sign.

⚠️ On the other hand, swapping the positions of the left and right sides of an inequality or multiplying/dividing both sides by a negative number requires you to reverse the position of the inequality sign such that: > becomes <, < becomes > , ≥ becomes ≤ , and ≤ becomes ≥

When solving inequalities with fractions or inequalities with variables on both sides, the process of using inverse operations to isolate the variable and solve remains the same, albeit with a few more steps involved.

Keep Learning:

Free Guide: How to Solve Compound Inequalities in 3 Easy Steps

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
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Equations and Inequalities Involving Signed Numbers

In chapter 2 we established rules for solving equations using the numbers of arithmetic. Now that we have learned the operations on signed numbers, we will use those same rules to solve equations that involve negative numbers. We will also study techniques for solving and graphing inequalities having one unknown.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

Upon completing this section you should be able to solve equations involving signed numbers.

Example 1 Solve for x and check: x + 5 = 3

Using the same procedures learned in chapter 2, we subtract 5 from each side of the equation obtaining

Example 2 Solve for x and check: - 3x = 12

Dividing each side by -3, we obtain

LITERAL EQUATIONS

• Identify a literal equation.
• Apply previously learned rules to solve literal equations.

An equation having more than one letter is sometimes called a literal equation . It is occasionally necessary to solve such an equation for one of the letters in terms of the others. The step-by-step procedure discussed and used in chapter 2 is still valid after any grouping symbols are removed.

Example 1 Solve for c: 3(x + c) - 4y = 2x - 5c

First remove parentheses.

At this point we note that since we are solving for c, we want to obtain c on one side and all other terms on the other side of the equation. Thus we obtain

Sometimes the form of an answer can be changed. In this example we could multiply both numerator and denominator of the answer by (- l) (this does not change the value of the answer) and obtain

The advantage of this last expression over the first is that there are not so many negative signs in the answer.

The most commonly used literal expressions are formulas from geometry, physics, business, electronics, and so forth.

Notice in this example that r was left on the right side and thus the computation was simpler. We can rewrite the answer another way if we wish.

GRAPHING INEQUALITIES

• Use the inequality symbol to represent the relative positions of two numbers on the number line.
• Graph inequalities on the number line.

The symbols are inequality symbols or order relations and are used to show the relative sizes of the values of two numbers. We usually read the symbol as "greater than." For instance, a > b is read as "a is greater than b." Notice that we have stated that we usually read a < b as a is less than b. But this is only because we read from left to right. In other words, "a is less than b" is the same as saying "b is greater than a." Actually then, we have one symbol that is written two ways only for convenience of reading. One way to remember the meaning of the symbol is that the pointed end is toward the lesser of the two numbers.

In simpler words this definition states that a is less than b if we must add something to a to get b. Of course, the "something" must be positive.

If you think of the number line, you know that adding a positive number is equivalent to moving to the right on the number line. This gives rise to the following alternative definition, which may be easier to visualize.

Example 1 3 < 6, because 3 is to the left of 6 on the number line.

Example 2 - 4 < 0, because -4 is to the left of 0 on the number line.

Example 3 4 > - 2, because 4 is to the right of -2 on the number line.

Example 4 - 6 < - 2, because -6 is to the left of -2 on the number line.

The mathematical statement x < 3, read as "x is less than 3," indicates that the variable x can be any number less than (or to the left of) 3. Remember, we are considering the real numbers and not just integers, so do not think of the values of x for x < 3 as only 2, 1,0, - 1, and so on.

As a matter of fact, to name the number x that is the largest number less than 3 is an impossible task. It can be indicated on the number line, however. To do this we need a symbol to represent the meaning of a statement such as x < 3.

The symbols ( and ) used on the number line indicate that the endpoint is not included in the set.

Example 5 Graph x < 3 on the number line.

Note that the graph has an arrow indicating that the line continues without end to the left.

Example 6 Graph x > 4 on the number line.

Example 7 Graph x > -5 on the number line.

Example 8 Make a number line graph showing that x > - 1 and x < 5. (The word "and" means that both conditions must apply.)

Example 9 Graph - 3 < x < 3.

Example 10 x >; 4 indicates the number 4 and all real numbers to the right of 4 on the number line.

The symbols [ and ] used on the number line indicate that the endpoint is included in the set.

Example 13 Write an algebraic statement represented by the following graph.

Example 14 Write an algebraic statement for the following graph.

Example 15 Write an algebraic statement for the following graph.

SOLVING INEQUALITIES

Upon completing this section you should be able to solve inequalities involving one unknown.

The solutions for inequalities generally involve the same basic rules as equations. There is one exception, which we will soon discover. The first rule, however, is similar to that used in solving equations.

If the same quantity is added to each side of an inequality , the results are unequal in the same order.

Example 1 If 5 < 8, then 5 + 2 < 8 + 2.

Example 2 If 7 < 10, then 7 - 3 < 10 - 3.

We can use this rule to solve certain inequalities.

Example 3 Solve for x: x + 6 < 10

If we add -6 to each side, we obtain

Graphing this solution on the number line, we have

We will now use the addition rule to illustrate an important concept concerning multiplication or division of inequalities.

Suppose x > a.

Now add - x to both sides by the addition rule.

Now add -a to both sides.

The last statement, - a > -x, can be rewritten as - x < -a. Therefore we can say, "If x > a, then - x < -a. This translates into the following rule:

If an inequality is multiplied or divided by a negative number, the results will be unequal in the opposite order.

Example 5 Solve for x and graph the solution: -2x>6

To obtain x on the left side we must divide each term by - 2. Notice that since we are dividing by a negative number, we must change the direction of the inequality.

Take special note of this fact. Each time you divide or multiply by a negative number, you must change the direction of the inequality symbol. This is the only difference between solving equations and solving inequalities.

Once we have removed parentheses and have only individual terms in an expression, the procedure for finding a solution is almost like that in chapter 2.

Let us now review the step-by-step method from chapter 2 and note the difference when solving inequalities.

First Eliminate fractions by multiplying all terms by the least common denominator of all fractions. (No change when we are multiplying by a positive number.) Second Simplify by combining like terms on each side of the inequality. (No change) Third Add or subtract quantities to obtain the unknown on one side and the numbers on the other. (No change) Fourth Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. (This is the important difference between equations and inequalities.)

• A literal equation is an equation involving more than one letter.
• The symbols are inequality symbols or order relations .
• a a is to the left of b on the real number line.
• To solve a literal equation for one letter in terms of the others follow the same steps as in chapter 2.
• To solve an inequality use the following steps: Step 1 Eliminate fractions by multiplying all terms by the least common denominator of all fractions. Step 2 Simplify by combining like terms on each side of the inequality. Step 3 Add or subtract quantities to obtain the unknown on one side and the numbers on the other. Step 4 Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. Step 5 Check your answer.

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• What are the 4 inequalities?
• There are four types of inequalities: greater than, less than, greater than or equal to, and less than or equal to.
• What is a inequality in math?
• In math, inequality represents the relative size or order of two values.
• How do you solve inequalities?
• To solve inequalities, isolate the variable on one side of the inequality, If you multiply or divide both sides by a negative number, flip the direction of the inequality.
• What are the 2 rules of inequalities?
• The two rules of inequalities are: If the same quantity is added to or subtracted from both sides of an inequality, the inequality remains true. If both sides of an inequality are multiplied or divided by the same positive quantity, the inequality remains true. If we multiply or divide both sides of an inequality by the same negative number, we must flip the direction of the inequality to maintain its truth.

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Inequalities

In Mathematics, equations are not always about being balanced on both sides with an 'equal to' symbol. Sometimes it can be about 'not an equal to' relationship like something is greater than the other or less than. In mathematics, inequality refers to a relationship that makes a non-equal comparison between two numbers or other mathematical expressions. These mathematical expressions come under algebra and are called inequalities.

Let us learn the rules of inequalities, and how to solve and graph them.

What is an Inequality?

Inequalities  are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

Olivia is selected in the 12U Softball. How old is Olivia? You don't know the age of Olivia, because it doesn't say "equals". But you do know her age should be less than or equal to 12, so it can be written as Olivia's Age ≤ 12. This is a practical scenario related to inequalities.

Inequality Meaning

The meaning of inequality is to say that two things are NOT equal. One of the things may be less than, greater than, less than or equal to, or greater than or equal to the other things.

• p ≠ q means that p is not equal to q
• p < q means that p is less than q
• p > q means that p is greater than q
• p ≤ q means that p is less than or equal to q
• p ≥ q means that p is greater than or equal to q

There are different types of inequalities. Some of the important inequalities are:

• Polynomial inequalities
• Absolute value inequalities
• Rational inequalities

Rules of Inequalities

The rules of inequalities are special. Here are some listed with inequalities examples.

Inequalities Rule 1

When inequalities are linked up you can jump over the middle inequality.

• If, p < q and q < d, then p < d
• If, p > q and q > d, then p > d

Example: If Oggy is older than Mia and Mia is older than Cherry, then Oggy must be older than Cherry.

Inequalities Rule 2

Swapping of numbers p and q results in:

• If, p > q, then q < p
• If, p < q, then q > p

Example: Oggy is older than Mia, so Mia is younger than Oggy.

Inequalities Rule 3

Adding the number d to both sides of inequality: If p < q, then p + d < q + d

Example: Oggy has less money than Mia. If both Oggy and Mia get $5 more, then Oggy will still have less money than Mia.

• If p < q, then p − d < q − d
• If p > q, then p + d > q + d, and
• If p > q, then p − d > q − d

So, the addition and subtraction of the same value to both p and q will not change the inequality.

Inequalities Rule 4

If you multiply numbers p and q by a positive number , there is no change in inequality. If you multiply both p and q by a negative number , the inequality swaps: p<q becomes q<p after multiplying by (-2)

Here are the rules:

• If p < q, and d is positive, then pd < qd
• If p < q, and d is negative, then pd > qd (inequality swaps)

Positive case example: Oggy's score of 5 is lower than Mia's score of 9 (p < q). If Oggy and Mia double their scores '×2', Oggy's score will still be lower than Mia's score, 2p < 2q. If the scores turn minuses, then scores will be −p > −q.

Inequalities Rule 5

Putting minuses in front of p and q changes the direction of the inequality.

• If p < q then −p > −q
• If p > q, then −p < −q
• It is the same as multiplying by (-1) and changes direction.

Inequalities Rule 6

Taking the reciprocal 1/value of both p and q changes the direction of the inequality. When p and q are both positive or both negative:

• If, p < q, then 1/p > 1/q
• If p > q, then 1/p < 1/q

Inequalities Rule 7

A square of a number is always greater than or equal to zero p 2  ≥ 0.

Example: (4) 2 = 16, (−4) 2 = 16, (0) 2 = 0

Inequalities Rule 8

Taking a square root will not change the inequality. If p ≤ q, then √p ≤ √q (for p, q ≥ 0).

Example: p=2, q=7 2 ≤ 7, then √2 ≤ √7

The rules of inequalities are summarized in the following table.

Solving Inequalities

Here are the steps for  solving inequalities :

• Step - 1: Write the inequality as an equation.
• Step - 2: Solve the equation for one or more values.
• Step - 3: Represent all the values on the number line.
• Step - 4: Also, represent all excluded values on the number line using open circles.
• Step - 5: Identify the intervals.
• Step - 6: Take a random number from each interval, substitute it in the inequality and check whether the inequality is satisfied.
• Step - 7: Intervals that are satisfied are the solutions.

But for solving simple inequalities (linear), we usually apply algebraic operations like addition , subtraction , multiplication , and division . Consider the following example:

2x + 3 > 3x + 4

Subtracting 3x and 3 from both sides,

2x - 3x > 4 - 3

Multiplying both sides by -1,

Notice that we have changed the ">" symbol into "<" symbol. Why? This is because we have multiplied both sides of the inequality by a negative number. The process of solving inequalities mentioned above works for a simple linear inequality. But to solve any other complex inequality, we have to use the following process.

Let us use this procedure to solve inequalities of different types.

Graphing Inequalities

While graphing inequalities , we have to keep the following things in mind.

• If the endpoint is included (i.e., in case of ≤ or ≥) use a closed circle.
• If the endpoint is NOT included (i.e., in case of < or >), use an open circle.
• Use open circle at either ∞ or -∞.
• Draw a line from the endpoint that extends to the right side if the variable is greater than the number.
• Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.

Writing Inequalities in Interval Notation

While writing the solution of an inequality in the interval notation , we have to keep the following things in mind.

• If the endpoint is included (i.e., in case of ≤ or ≥) use the closed brackets '[' or ']'
• If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
• Use always open bracket at either ∞ or -∞.

Here are some examples to understand the same:

Graphing Inequalities with Two Variables

For graphing  inequalities with two variables , you will have to plot the "equals" line and then, shade the appropriate area. There are three steps:

• Write the equation such as "y" is on the left and everything else on the right.
• Plot the "y=" line (draw a solid line for y≤ or y≥, and a dashed line for y< or y>)
• Shade the region above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).

Let us try some example: This is a graph of a linear inequality: y ≤ x + 4

You can see, y = x + 4 line and the shaded area (in yellow) is where y is less than or equal to x + 4. Let us now see how to solve different types of inequalities and how to graph the solution in each case.

Solving Polynomial Inequalities

The polynomial inequalities are inequalities that can be expressed as a polynomial on one side and 0 on the other side of the inequality. There are different types of polynomial inequalities but the important ones are:

• Linear Inequalities
• Quadratic Inequalities

Solving Linear Inequalities

A linear inequality is an inequality that can be expressed with a linear expression on one side and a 0 on the other side. Solving linear inequalities is as same as solving linear equations , but just the rules of solving inequalities (that was explained before) should be taken care of. Let us see some examples.

Solving One Step Inequalities

Consider an inequality 2x < 6 (which is a linear inequality with one variable ). To solve this, just one step is sufficient which is dividing both sides by 2. Then we get x < 3. Therefore, the solution of the inequality is x < 3 (or) (-∞, 3).

Solving Two Step Inequalities

Consider an inequality -2x + 3 > 6. To solve this, we need two steps . The first step is subtracting 3 from both sides, which gives -2x > 3. Then we need to divide both sides by -2 and it results in x < -3/2 (note that we have changed the sign of the inequality). So the solution of the inequality is x < -3/2 (or) (-∞, -3/2).

Solving Compound Inequalities

Compound inequalities refer to the set of inequalities with either "and" or "or" in between them. For solving inequalities, in this case, just solve each inequality independently and then find the final solution according to the following rules:

• The final solution is the intersection of the solutions of the independent inequalities if we have "and" between them.
• The final solution is the union of the solutions of the independent inequalities if we have "or" between them.

Example: Solve the compound inequality 2x + 3 < -5 and x + 6 < 3.

By first inequality: 2x + 3 < -5 2x < -8 x < -4

By second inequality, x + 6 < 3 x < -3

Since we have "and" between them, we have to find the intersection of the sets x < -4 and x < -3. A number line may be helpful in this case. Then the final solution is:

x < -3 (or) (-∞, -3).

Solving Quadratic Inequalities

A quadratic inequality involves a quadratic expression in it. Here is the process of solving quadratic inequalities . The process is explained with an example where we are going to solve the inequality x 2 - 4x - 5 ≥ 0.

• Step 1: Write the inequality as equation. x 2 - 4x - 5 = 0
• Step 2: Solve the equation. Here we can use any process of solving quadratic equations . Then (x - 5) (x + 1) = 0 x = 5; x = -1.

• Step 5: The inequalities with "true" from the above table are solutions. Therefore, the solutions of the quadratic inequality x 2 - 4x - 5 ≥ 0 is (-∞, -1] U [5, ∞).

We can use the same process for solving cubic inequalities, biquadratic inequalities, etc.

Solving Absolute Value Inequalities

An absolute value inequality includes an algebraic expression inside the absolute value sign. Here is the process of solving absolute value inequalities where the process is explained with an example of solving an absolute value inequality |x + 3| ≤ 2. If you want to learn different methods of solving absolute value inequalities, click here .

• Step 1: Consider the absolute value inequality as equation. |x + 3| = 2
• Step 2: Solve the equation. x + 3 = ±2 x + 3 = 2; x + 3 = -2 x = -1; x = -5

• Step 5: The intervals that satisfied the inequality are the solution intervals. Therefore, the solution of the absolute value inequality |x + 3| ≤ 2 is [-5, -1].

Solving Rational Inequalities

Rational inequalities are inequalities that involve rational expressions (fractions with variables). To solve the rational inequalities (inequalities with fractions), we just use the same procedure as other inequalities but we have to take care of the excluded points . For example, while solving the rational inequality (x + 2) / (x - 2) < 3, we should note that the rational expression (x + 2) / (x - 2) is NOT defined at x = 2 (set the denominator x - 2 = 0 ⇒x = 2). Let us solve this inequality step by step.

• Step 1: Consider the inequality as the equation. (x + 2) / (x - 2) = 3
• Step 2: Solve the equation. x + 2 = 3(x - 2) x + 2 = 3x - 6 2x = 8 x = 4

• Step 5: The intervals that have come up with "true" in Step 4 are the solutions. Therefore, the solution of the rational inequality (x + 2) / (x - 2) < 3 is (-∞, 2) U (4, ∞).

Important Notes on Inequalities:

Here are the notes about inequalities:

• If we have strictly less than or strictly greater than symbol, then we never get any closed interval in the solution.
• We always get open intervals at ∞ or -∞ symbols because they are NOT numbers to include.
• Write open intervals always at excluded values when solving rational inequalities.
• Excluded values should be taken care of only in case of rational inequalities.

☛  Related Topics:

• Linear Inequalities Calculator
• Triangle Inequality Theorem Calculator
• Rational Inequalities Calculator

Inequalities Examples

Example 1: Using the techniques of solving inequalities, solve: -19 < 3x + 2 ≤ 17 and write the answer in the interval notation.

Given that -19 < 3x + 2 ≤ 17.

This is a compound inequality.

Subtracting 2 from all sides,

-21 < 3x ≤ 15

Dividing all sides by 3,

-7 < x ≤ 5

Answer: The solution is (-7, 5].

Example 2: While solving inequalities, explain why each of the following statements is incorrect. Also, correct them. a) 2x < 5 ⇒ x > 5/2 b) x > 3 ⇒ x ∈ [3, ∞) c) -x > -7 ⇒ x > 7.

a) 2x < 5. Here, when we divide both sides by 2, which is a positive number, the sign does not change. So the correct inequality is x < 5/2.

b) x > 3. It does not include an equal to symbol. So 3 should NOT be included in the interval. So the correct interval is (3, ∞).

c) -x > -7. When we divide both sides by -1, a negative number, the sign should change. So the correct inequality is x < 7.

Answer: The corrected ones are a) x < 5/2; b) x ∈ (3, ∞); c) x < 7.

Example 3: Solve the inequality x 2 - 7x + 10 < 0.

First, solve the equation x 2 - 7x + 10 = 0.

(x - 2) (x - 5) = 0.

x = 2, x = 5.

If we represent these numbers on the number line, we get the following intervals: (-∞, 2), (2, 5), and (5, ∞).

Let us take some random numbers from each interval to test the given quadratic inequality.

Therefore, the only interval that satisfies the inequality is (2, 5).

Answer: The solution is (2, 5).

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Practice Questions on Inequalities

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FAQs on Inequalities

What are inequalities in math.

When two or more algebraic expressions are compared using the symbols <, > ≤, or ≥, then they form an inequality. They   are the mathematical expressions in which both sides are not equal.

How Do you Solve Inequalities On A Number Line?

To plot an inequality in math, such as x>3, on a number line,

• Step 1: Draw a circle over the number (e.g., 3).
• Step 2: Check if the sign includes equal to (≥ or ≤) or not. If equal to sign is there along with > or <, then fill in the circle otherwise leave the circle unfilled.
• Step 3: On the number line, extend the line from 3(after encircling it) to show it is greater than or equal to 3.

How to Calculate Inequalities in Math?

To  calculate inequalities :

• just make it an equation
• mark the zeros  on the number line to get intervals
• test the intervals by taking any one number from it against the inequality.

Explain the Process of Solving Inequalities Graphically.

Solving inequalities graphically is possible when we have a system of two inequalities in two variables. In this case, we consider both inequalities as two linear equations and graph them. Then we get two lines. Shade the upper/lower portion of each of the lines that satisfies the inequality. The common portion of both shaded regions is the solution region.

What is the Difference Between Equations and Inequalities?

Here are the differences between equations and inequalities.

What Happens When you Square An Inequality?

A square of a number is always greater than or equal to zero p 2  ≥ 0. Example: (4) 2 = 16, (−4) 2 = 16, (0) 2 = 0

What are the Steps to Calculate Inequalities with Fractions?

Calculating inequalities with fractions is just like solving any other inequality. One easy way of solving such inequalities is to multiply every term on both sides by the LCD of all denominators so that all fractions become integers . For example, to solve (1/2) x + 1 > (3/4) x + 2, multiply both sides by 4. Then we get 2x + 4 > 3x + 8 ⇒ -x > 4 ⇒ x < -4.

What are the Steps for Solving Inequalities with Variables on Both Sides?

When an inequality has a variable on both sides, we have to try to isolate the variable. But in this process, flip the inequality sign whenever we are dividing or multiplying both sides by a negative number. Here is an example. 3x - 7 < 5x - 11 ⇒ -2x < -4 ⇒ x > 2.

How Do you Find the Range of Inequality?

You can find the range of values of x, by solving the inequality by considering it as a normal linear equation.

What Are the 5 Inequality Symbols?

The 5 inequality symbols are less than (<), greater than (>), less than or equal (≤), greater than or equal (≥), and the not equal symbol (≠).

How Do you Tell If It's An Inequality?

Equations and inequalities are mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are supposed to be equal and shown by the symbol =. Whereas in inequality, the two expressions are not necessarily equal and are indicated by the symbols: >, <, ≤ or ≥.

How to Graph the Solution After Solving Inequalities?

After solving inequalities, we can graph the solution keeping the following things in mind.

• Use an open circle at the number if it is not included and use a closed circle if it is included.
• Draw a line to the right side of the number in case of '>' and to the left side of the number in case of '<'.

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2.7: Solve Linear Inequalities

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Learning Objectives

By the end of this section, you will be able to:

• Graph inequalities on the number line
• Solve inequalities using the Subtraction and Addition Properties of inequality
• Solve inequalities using the Division and Multiplication Properties of inequality
• Solve inequalities that require simplification
• Translate to an inequality and solve

Before you get started, take this readiness quiz.

• Translate from algebra to English: \(15>x\). If you missed this problem, review Exercise 1.3.1 .
• Solve: \(n−9=−42\). If you missed this problem, review Exercise 2.1.7 .
• Solve: \(−5p=−23\). If you missed this problem, review Exercise 2.2.1 .
• Solve: \(3a−12=7a−20\). If you missed this problem, review Exercise 2.3.22 .

Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality \(x > 3\) true? Are you thinking, ‘ x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality \(x > 3\).

We show the solutions to the inequality \(x > 3\) on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of \(x > 3\) is shown in Figure \(\PageIndex{1}\). Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.

The graph of the inequality \(x \geq 3\) is very much like the graph of \(x > 3\), but now we need to show that 3 is a solution, too. We do that by putting a bracket at \(x = 3\), as shown in Figure \(\PageIndex{2}\).

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

Exercise \(\PageIndex{1}\)

Graph on the number line:

• \(x\leq 1\)
• \(x>−1\)

Exercise \(\PageIndex{2}\)

• \(x\leq −1\)

Exercise \(\PageIndex{3}\)

• \(x>−2\)
• \(x<−3\)
• \(x\geq −1\)

We can also represent inequalities using interval notation. As we saw above, the inequality \(x>3\) means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation , we express \(x>3\) as \((3, \infty)\). The symbol \(\infty\) is read as ‘infinity’. It is not an actual number. Figure \(\PageIndex{3}\) shows both the number line and the interval notation.

The inequality \(x\leq 1\) means all numbers less than or equal to 1. There is no lower end to those numbers. We write \(x\leq 1\) in interval notation as \((-\infty, 1]\). The symbol \(-\infty\) is read as ‘negative infinity’. Figure \(\PageIndex{4}\) shows both the number line and interval notation.

INEQUALITIES, NUMBER LINES, AND INTERVAL NOTATION

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure \(\PageIndex{5}\).

Exercise \(\PageIndex{4}\)

Graph on the number line and write in interval notation.

• \(x \geq -3\)
• \(x<2.5\)
• \(x\leq \frac{3}{5}\)

Exercise \(\PageIndex{5}\)

Graph on the number line and write in interval notation:

• \(x\leq −1.5\)
• \(x\geq \frac{3}{4}\)

Exercise \(\PageIndex{6}\)

• \(x\leq −4\)
• \(x\geq 0.5\)
• \(x<-\frac{2}{3}\)

Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

PROPERTIES OF EQUALITY

\[\begin{array} { l l } { \textbf { Subtraction Property of Equality } } & { \textbf { Addition Property of Equality } } \\ { \text { For any numbers } a , b , \text { and } c , } & { \text { For any numbers } a , b , \text { and } c } \\ { \text { if } \qquad \quad a = b , } & { \text { if } \qquad \quad a = b } \\ { \text { then } a - c = b - c . } & { \text { then } a + c = b + c } \end{array}\]

Similar properties hold true for inequalities.

Similarly we could show that the inequality also stays the same for addition.

This leads us to the Subtraction and Addition Properties of Inequality.

PROPERTIES OF INEQUALITY

\[\begin{array} { l l } { \textbf { Subtraction Property of Inequality } } & { \textbf { Addition Property of Inequality } } \\ { \text { For any numbers } a , b , \text { and } c , } & { \text { For any numbers } a , b , \text { and } c } \\ { \text { if }\qquad \quad a < b } & { \text { if } \qquad \quad a < b } \\ { \text { then } a - c < b - c . } & { \text { then } a + c < b + c } \\\\ { \text { if } \qquad \quad a > b } & { \text { if } \qquad \quad a > b } \\ { \text { then } a - c > b - c . } & { \text { then } a + c > b + c } \end{array}\]

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality \(x+5>9\), the steps would look like this:

\[\begin{array}{rrll} {} &{x + 5} &{ >} &{9} \\ {\text{Subtract 5 from both sides to isolate }x.} &{x + 5 - 5} &{ >} &{9 - 5} \\{} &{x} &{ >} &{4} \\ \end{array}\]

Any number greater than 4 is a solution to this inequality.

Exercise \(\PageIndex{7}\)

Solve the inequality \(n - \frac{1}{2} \leq \frac{5}{8}\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{8}\)

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

\(p - \frac{3}{4} \geq \frac{1}{6}\)

Exercise \(\PageIndex{9}\)

\(r - \frac{1}{3} \leq \frac{7}{12}\)

Solve Inequalities using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

\[\begin{array}{ll} {\textbf{Division Property of Equality}} &{\textbf{MUltiplication Property of Equality}} \\ {\text{For any numbers a, b, c, and c} \neq 0} &{\text{For any numbers a, b, c}} \\ {\text{if } \qquad a = b} &{\text{if} \qquad \quad a = b} \\ {\text{then }\quad \frac{a}{c} = \frac{b}{c}} &{\text{then } \quad ac = bc} \end{array}\]

Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

Consider some numerical examples.

The inequality signs stayed the same.

Does the inequality stay the same when we divide or multiply by a negative number?

The inequality signs reversed their direction.

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

Here are the Division and Multiplication Properties of Inequality for easy reference.

DIVISION AND MULTIPLICATION PROPERTIES OF INEQUALITY

For any real numbers a,b,c

\[\begin{array}{ll} {\text{if } a < b \text{ and } c > 0, \text{ then}} &{\frac{a}{c} < \frac{b}{c} \text{ and } ac < bc} \\ {\text{if } a > b \text{ and } c > 0, \text{ then}} &{\frac{a}{c} > \frac{b}{c} \text{ and } ac > bc} \\ {\text{if } a < b \text{ and } c < 0, \text{ then}} &{\frac{a}{c} > \frac{b}{c} \text{ and } ac > bc} \\ {\text{if } a > b \text{ and } c < 0, \text{ then}} &{\frac{a}{c} < \frac{b}{c} \text{ and } ac < bc} \end{array}\]

When we divide or multiply an inequality by a:

• positive number, the inequality stays the same .
• negative number, the inequality reverses .

Exercise \(\PageIndex{10}\)

Solve the inequality \(7y<​​42\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{11}\)

\(9c>72\)

\((8, \infty)\)

Exercise \(\PageIndex{12}\)

\(12d\leq 60\)

\(d\leq 5\)

\((-\infty, 5]\)

Exercise \(\PageIndex{13}\)

Solve the inequality \(−10a\geq 50\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{14}\)

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

\(−8q<32\)

\(q>−4\)

Exercise \(\PageIndex{15}\)

\(−7r\leq −70\)

SOLVING INEQUALITIES

\[\begin{array}{l} x > a\text{ has the same meaning as } a < x \end{array}\]

Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”

Exercise \(\PageIndex{16}\)

Solve the inequality \(-20 < \frac{4}{5}u\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{17}\)

\(24 \leq \frac{3}{8}m\)

Exercise \(\PageIndex{18}\)

\(-24 < \frac{4}{3}n\)

Exercise \(\PageIndex{19}\)

Solve the inequality \(\frac{t}{-2} \geq 8\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{20}\)

\(\frac{k}{-12}\leq 15\)

Exercise \(\PageIndex{21}\)

\(\frac{u}{-4}\geq -16\)

Solve Inequalities That Require Simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

Exercise \(\PageIndex{22}\)

Solve the inequality \(4m\leq 9m+17\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{23}\)

Solve the inequality \(3q\geq 7q−23\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{24}\)

Solve the inequality \(6x<10x+19\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{25}\)

Solve the inequality \(8p+3(p−12)>7p−28\) graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{26}\)

Solve the inequality \(9y+2(y+6)>5y−24\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{27}\)

Solve the inequality \(6u+8(u−1)>10u+32\), graph the solution on the number line, and write the solution in interval notation.

Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

Exercise \(\PageIndex{28}\)

Solve the inequality \(8x−2(5−x)<4(x+9)+6x\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{29}\)

Solve the inequality \(4b−3(3−b)>5(b−6)+2b\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{30}\)

Solve the inequality \(9h−7(2−h)<8(h+11)+8h\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{31}\)

Solve the inequality \(\frac{1}{3}a - \frac{1}{8}a > \frac{5}{24}a + \frac{3}{4}\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{32}\)

Solve the inequality \(\frac{1}{4}x - \frac{1}{12}x > \frac{1}{6}x + \frac{7}{8}\), graph the solution on the number line, and write the solution in interval notation.

Exercise \(\PageIndex{33}\)

Solve the inequality \(\frac{2}{5}z - \frac{1}{3}z < \frac{1}{15}z - \frac{3}{5}\), graph the solution on the number line, and write the solution in interval notation.

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.

Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.

Table \(\PageIndex{4}\) ​ shows some common phrases that indicate inequalities.

Exercise \(\PageIndex{34}\)

Translate and solve. Then write the solution in interval notation and graph on the number line.

Twelve times c is no more than 96.

Exercise \(\PageIndex{35}\)

Twenty times y is at most 100

Exercise \(\PageIndex{36}\)

Nine times z is no less than 135

Exercise \(\PageIndex{37}\)

Thirty less than x is at least 45.

Exercise \(\PageIndex{38}\)

Nineteen less than p is no less than 47

Exercise \(\PageIndex{39}\)

Four more than a is at most 15.

Key Concepts

• Subtraction Property of Inequality For any numbers a, b, and c, if a<b then a−c<b−c and if a>b then a−c>b−c.
• Addition Property of Inequality For any numbers a, b, and c, if a<b then a+c<b+c and if a>b then a+c>b+c.
• Division and Multiplication Properties of Inequalit y For any numbers a, b, and c, if a<b and c>0, then ac<bc and ac>bc. if a>b and c>0, then ac>bc and ac>bc. if a<b and c<0, then ac>bc and ac>bc. if a>b and c<0, then ac<bc and ac<bc.

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Section 2.11 : Linear Inequalities

To this point in this chapter we’ve concentrated on solving equations. It is now time to switch gears a little and start thinking about solving inequalities. Before we get into solving inequalities we should go over a couple of the basics first.

At this stage of your mathematical career it is assumed that you know that

means that \(a\) is some number that is strictly less than \(b\). It is also assumed that you know that

means that \(a\) is some number that is either strictly bigger than \(b\) or is exactly equal to \(b\). Likewise, it is assumed that you know how to deal with the remaining two inequalities. > (greater than) and \( \le \) (less than or equal to).

What we want to discuss is some notational issues and some subtleties that sometimes get students when they really start working with inequalities.

First, remember that when we say that \(a\) is less than \(b\) we mean that \(a\) is to the left of \(b\) on a number line. So,

is a true inequality.

Next, don’t forget how to correctly interpret \( \le \) and \( \ge \). Both of the following are true inequalities.

In the first case 4 is equal to 4 and so it is “less than or equal” to 4. In the second case -6 is strictly less than 4 and so it is “less than or equal” to 4. The most common mistake is to decide that the first inequality is not a true inequality. Also be careful to not take this interpretation and translate it to < and/or >. For instance,

is not a true inequality since 4 is equal to 4 and not strictly less than 4.

Finally, we will be seeing many double inequalities throughout this section and later sections so we can’t forget about those. The following is a double inequality.

In a double inequality we are saying that both inequalities must be simultaneously true. In this case 5 is definitely greater than -9 and at the same time is less than or equal to 6. Therefore, this double inequality is a true inequality.

On the other hand,

is not a true inequality. While it is true that 5 is less than 20 (so the second inequality is true) it is not true that 5 is greater than or equal to 10 (so the first inequality is not true). If even one of the inequalities in a double inequality is not true then the whole inequality is not true. This point is more important than you might realize at this point. In a later section we will run across situations where many students try to combine two inequalities into a double inequality that simply can’t be combined, so be careful.

The next topic that we need to discuss is the idea of interval notation . Interval notation is some very nice shorthand for inequalities and will be used extensively in the next few sections of this chapter.

The best way to define interval notation is the following table. There are three columns to the table. Each row contains an inequality, a graph representing the inequality and finally the interval notation for the given inequality.

Remember that a bracket, “[” or “]”, means that we include the endpoint while a parenthesis, “(” or “)”, means we don’t include the endpoint.

Now, with the first four inequalities in the table the interval notation is really nothing more than the graph without the number line on it. With the final four inequalities the interval notation is almost the graph, except we need to add in an appropriate infinity to make sure we get the correct portion of the number line. Also note that infinities NEVER get a bracket. They only get a parenthesis.

We need to give one final note on interval notation before moving on to solving inequalities. Always remember that when we are writing down an interval notation for an inequality that the number on the left must be the smaller of the two.

It’s now time to start thinking about solving linear inequalities. We will use the following set of facts in our solving of inequalities. Note that the facts are given for <. We can however, write down an equivalent set of facts for the remaining three inequalities.

• If \(a < b\) then \(a + c < b + c\) and \(a - c < b - c\)for any number \(c\). In other words, we can add or subtract a number to both sides of the inequality and we don’t change the inequality itself.
• If \(a < b\) and \(c > 0\) then \(ac < bc\) and \(\frac{a}{c} < \frac{b}{c}\). So, provided \(c\) is a positive number we can multiply or divide both sides of an inequality by the number without changing the inequality.
• If \(a < b\) and \(c < 0\) then \(ac > bc\) and \(\frac{a}{c} > \frac{b}{c}\). In this case, unlike the previous fact, if \(c\) is negative we need to flip the direction of the inequality when we multiply or divide both sides by the inequality by \(c\).

These are nearly the same facts that we used to solve linear equations. The only real exception is the third fact. This is the important fact as it is often the most misused and/or forgotten fact in solving inequalities.

If you aren’t sure that you believe that the sign of \(c\) matters for the second and third fact consider the following number example.

I hope that we would all agree that this is a true inequality. Now multiply both sides by 2 and by -2.

Sure enough, when multiplying by a positive number the direction of the inequality remains the same, however when multiplying by a negative number the direction of the inequality does change.

Okay, let’s solve some inequalities. We will start off with inequalities that only have a single inequality in them. In other words, we’ll hold off on solving double inequalities for the next set of examples.

The thing that we’ve got to remember here is that we’re asking to determine all the values of the variable that we can substitute into the inequality and get a true inequality. This means that our solutions will, in most cases, be inequalities themselves.

• \( - 2\left( {m - 3} \right) < 5\left( {m + 1} \right) - 12\)
• \(2\left( {1 - x} \right) + 5 \le 3\left( {2x - 1} \right)\)

Solving single linear inequalities follow pretty much the same process for solving linear equations. We will simplify both sides, get all the terms with the variable on one side and the numbers on the other side, and then multiply/divide both sides by the coefficient of the variable to get the solution. The one thing that you’ve got to remember is that if you multiply/divide by a negative number then switch the direction of the inequality.

There really isn’t much to do here other than follow the process outlined above.

You did catch the fact that the direction of the inequality changed here didn’t you? We divided by a “-7” and so we had to change the direction. The inequality form of the solution is \(m > \frac{{13}}{7}\). The interval notation for this solution is, \(\left( {\frac{{13}}{7},\infty } \right)\).

Again, not much to do here.

Now, with this inequality we ended up with the variable on the right side when it more traditionally on the left side. So, let’s switch things around to get the variable onto the left side. Note however, that we’re going to need also switch the direction of the inequality to make sure that we don’t change the answer. So, here is the inequality notation for the inequality.

Now, let’s solve some double inequalities. The process here is similar in some ways to solving single inequalities and yet very different in other ways. Since there are two inequalities there isn’t any way to get the variables on “one side” of the inequality and the numbers on the other. It is easier to see how these work if we do an example or two so let’s do that.

• \( - 6 \le 2\left( {x - 5} \right) < 7\)
• \( - 3 < \frac{3}{2}\left( {2 - x} \right) \le 5\)
• \( - 14 < - 7\left( {3x + 2} \right) < 1\)

The process here is fairly similar to the process for single inequalities, but we will first need to be careful in a couple of places. Our first step in this case will be to clear any parenthesis in the middle term.

Now, we want the \(x\) all by itself in the middle term and only numbers in the two outer terms. To do this we will add/subtract/multiply/divide as needed. The only thing that we need to remember here is that if we do something to middle term we need to do the same thing to BOTH of the out terms. One of the more common mistakes at this point is to add something, for example, to the middle and only add it to one of the two sides.

Okay, we’ll add 10 to all three parts and then divide all three parts by two.

That is the inequality form of the answer. The interval notation form of the answer is \(\left[ {2,\frac{{17}}{2}} \right)\).

In this case the first thing that we need to do is clear fractions out by multiplying all three parts by 2. We will then proceed as we did in the first part.

Now, we’re not quite done here, but we need to be very careful with the next step. In this step we need to divide all three parts by -3. However, recall that whenever we divide both sides of an inequality by a negative number we need to switch the direction of the inequality. For us, this means that both of the inequalities will need to switch direction here.

So, there is the inequality form of the solution. We will need to be careful with the interval notation for the solution. First, the interval notation is NOT \(\left( {4, - \frac{4}{3}} \right]\). Remember that in interval notation the smaller number must always go on the left side! Therefore, the correct interval notation for the solution is \(\left[ { - \frac{4}{3},4} \right)\).

Note as well that this does match up with the inequality form of the solution as well. The inequality is telling us that \(x\) is any number between 4 and \( - \frac{4}{3}\) or possibly \( - \frac{4}{3}\) itself and this is exactly what the interval notation is telling us.

Also, the inequality could be flipped around to get the smaller number on the left if we’d like to. Here is that form,

When doing this make sure to correctly deal with the inequalities as well.

Not much to this one. We’ll proceed as we’ve done the previous two.

Don’t get excited about the fact that one of the sides is now zero. This isn’t a problem. Again, as with the last part, we’ll be dividing by a negative number and so don’t forget to switch the direction of the inequalities.

When solving double inequalities make sure to pay attention to the inequalities that are in the original problem. One of the more common mistakes here is to start with a problem in which one of the inequalities is < or > and the other is \( \le \) or \( \ge \), as we had in the first two parts of the previous example, and then by the final answer they are both < or > or they are both \( \le \) or \( \ge \). In other words, it is easy to all of a sudden make both of the inequalities the same. Be careful with this.

There is one final example that we want to work here.

This is easier than it may appear at first. All we are really going to do is start with the given inequality and then manipulate the middle term to look like the second inequality. Again, we’ll need to remember that whatever we do to the middle term we’ll also need to do to the two outer terms.

So, first we’ll multiply everything by 2.

Now add 3 to everything.

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Choosing a Method to Solve Quadratics - Video Tutorials & Practice Problems

Choosing a method to solve quadratics.

So you've just learned all possible methods of solving a quadratic equation. But that means that you can just be given a quadratic equation with no direction on what method to use and just be expected to choose the best one. But how do we know which method is best for any given quadratic? Well, we've been collecting information on when we should use each method. And here I'm going to use that information to show you exactly how to choose method is best. Now, it's important to note here that multiple methods may work for any given quadratic. But here we're going to be focused on choosing which one is best, which one is going to make our quadratic equation, the easiest and most straightforward to solve. So let's go ahead and get started. So in our table here, we're going to look at our use if column and a structured way to go through. This is to start with factoring and go go through each method until you find one that matches up with your quadratic equation. Now, as you get more practice, you're going to be able to just look at a quadratic and you might be able to see right away which method it matches up with and that's fine as well. But let's go ahead and start with some structure. So looking at our first example here, we want to choose which method is best and I have X squared plus three X equals zero. So taking a look first at my factoring box, let's check if any of this criteria matches. So it either has obvious factors or C is equal to zero. Now, here I do have C equal to zero because I have no constant. So I can actually stop here because I know that factoring is going to be the best method to use. So let's go ahead and take a look at our second example here I have X squared plus six X plus one is equal to zero. Let's again, start with our factoring box. So does this have obvious factors or is C equal to zero? Well, it doesn't have obvious factors to me and I do have a constant, I have this one. So factoring is not going to be the best choice here. So let's move on to the square root property. So does it have this form X plus some number squared equals a constant? No, and is B equal to zero. Also know here because I have a B of six. So I'm not gonna be able to use the square root property either let's move on to completing the square. So is my leading coefficient one and is B even. Well, I do have a leading coefficient of one because that X squared is just by itself there and B is six, which is an even number. So that tells me that completing the square is going to be the best choice to solve this one. So let's look at another example here. So we've been able to kind of go through each of them and see which one works. Let's take a look at one more. So here I have X plus two squared equals nine. Now this is an example of a quadratic that we're gonna be able to look at and know exactly what me to use because there's such a specific thing happening. So I have X plus A number squared is equal to a constant, which I know is one of the clues that I should use the square root property. So here the square root property is going to be the best method. So now we just have one more left. Let's take a look. So I have two X squared plus seven X plus three equals zero. Let's go ahead and go through a left to right one more time. So does this have obvious factors? Well, definitely not and C is not equal to zero, it's equal to three. So I'm not gonna want a factor here moving on to the square root property. It definitely doesn't have, have this form X plus a number squared equals a constant. And I do have a B because I have this seven X. So not the square root property either. And completing the square, is this going to be the best method? Well, my leading coefficient is not one. So the, the completing the square is already out of the picture which just leaves me with the quadratic formula. So that is going to be the best method here. So remember again, multiple methods can work for any quadratic, but these are the best ones. And that's how to choose the best method for any given quadratic equation. Let me know if you have any questions.

Choose and apply the best method to solve the given quadratic equation.  x 2 − 6 x = 5 x^2-6x=5 x 2 − 6 x = 5 

 x = 6 + 14 , x = 6 − 14 x=6+\sqrt{14},x=6-\sqrt{14} x = 6 + 14 ​ , x = 6 − 14 ​ 

 x = 10 , x = − 4 x=10,x=-4 x = 10 , x = − 4 

 x = 3 + 14 , x = 3 − 14 x=3+\sqrt{14},x=3-\sqrt{14} x = 3 + 14 ​ , x = 3 − 14 ​ 

 x = 6 , x = 0 x=6,x=0 x = 6 , x = 0 

Choose and apply the best method to solve the given quadratic equation.   4 x 2 + 16 x + 12 = 0 4x^2+16x+12=0 4 x 2 + 16 x + 12 = 0 

 x = − 3 , x = − 1 x=-3,x=-1 x = − 3 , x = − 1 

 x = 3 , x = 1 x=3,x=1 x = 3 , x = 1 

 x = 3 , x = 4 x=3,x=4 x = 3 , x = 4 

 x = − 12 , x = 1 x=-12,x=1 x = − 12 , x = 1 

Do you want more practice?

• Match the equation in Column I with its solution(s) in Column II. x^2 = 25
• Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer,...
• Solve each equation in Exercises 1 - 14 by factoring. 2x(x - 3) = 5x^2 - 7x
• Solve each equation in Exercises 1 - 14 by factoring. 10x - 1 = (2x + 1)^2
• Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x ...
• Solve each equation in Exercises 15–34 by the square root property. (x + 2)^2 = 25
• Solve each equation in Exercises 15–34 by the square root property. (x + 3)^2 = - 16
• Solve each equation in Exercises 15–34 by the square root property. (2x + 8)^2 = 27
• Solve each equation. -2x² +11x = -21
• Solve each equation using completing the square. See Examples 3 and 4. 2x^2 + x = 10
• Solve each equation in Exercises 47–64 by completing the square. x^2 + 6x = 7
• See Exercise 47. (b)Which equation has two nonreal complex solutions?
• Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 = 2x - 5
• Solve each equation in Exercises 47–64 by completing the square. 2x^2 - 7x + 3 = 0
• Solve each equation in Exercises 60–63 by the square root property. x^2/2 + 5 = -3
• Solve each equation in Exercises 66–67 by completing the square. 3x^2 -12x+11= 0
• Solve each equation in Exercises 65–74 using the quadratic formula. 4x^2 = 2x + 7
• Exercises 73–75 will help you prepare for the material covered in the next section. Multiply: (7 - 3x)(- 2 - ...
• In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given eq...
• Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and te...
• Solve each equation in Exercises 83–108 by the method of your choice. 5x^2 + 2 = 11x
• Answer each question. Find the values of a, b, and c for which the quadratic equation. ax^2 + bx + c = 0 has ...
• Solve each equation in Exercises 83–108 by the method of your choice. 9 - 6x + x^2 = 0
• Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 3) = 1/4
• Solve each equation in Exercises 83–108 by the method of your choice. 3/(x - 3) + 5/(x - 4) = (x^2 - 20)/(x^2...
• In Exercises 115–122, find all values of x satisfying the given conditions. y1 = 2x^2 + 5x - 4, y2 = - x^2 + ...
• Write a quadratic equation in general form whose solution set is {- 3, 5}.