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## 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:

## Example: x + 2 > 12

## 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 !

## 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):

## Example: 2y+7 < 12

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

## Adding or Subtracting a Value

## Example: x + 3 < 7

If we subtract 3 from both sides, we get:

And that is our solution: x < 4

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

## What did we do?

## 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:

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:

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 >) ?

## Example: −2y < −8

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

And that is the correct solution: y > 4

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

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 .

## A Bigger Example

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.

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:

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

- 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)

## Solving Inequalities

Related Pages Solving Equations Algebraic Expressions More Algebra Lessons

In these lessons, we will look at the rules, approaches, and techniques for solving inequalities.

To solve an inequality, we can:

- Add the same number to both sides.
- Subtract the same number from both sides.
- Multiply both sides by the same positive number.
- Divide both sides by the same positive number.
- Multiply both sides by the same negative number and reverse the sign.
- Divide both sides by the same negative number and reverse the sign.

## Inequalities Of The Form “x + a > b” or “x + a < b”

Solution: x + 7 < 15 x + 7 – 7 < 15 – 7 x < 8

## Inequalities Of The Form “x – a < b” or “x – a > b”

Solution: x – 6 > 14 x – 6+ 6 > 14 + 6 x > 20

Example: Solve the inequality x – 3 + 2 < 10

Solution: x – 3 + 2 < 10 x – 1 < 10 x – 1 + 1 < 10 + 1 x < 11

## Inequalities Of The Form “a – x < b” or “a – x > b”

Example: Solve the inequality 7 – x < 9

Example: Solve the inequality 12 > 18 – y

## Inequalities Of The Form “ < b” or “ > b”

Solving linear inequalities with like terms.

Example: Evaluate 3x – 8 + 2x < 12

Solution: 3x – 8 + 2x < 12 3x + 2x < 12 + 8 5x < 20 x < 4

Example: Evaluate 6x – 8 > x + 7

Solution: 6x – 8 > x + 7 6x – x > 7 + 8 5x > 15 x > 3

Example: Evaluate 2(8 – p) ≤ 3(p + 7)

## An Introduction To Solving Inequalities

## Solving One-Step Linear Inequalities In One Variable

## Solving Two-Step Linear Inequalities In One Variable

## Solving Linear Inequalities

Examples of how to solve linear inequalities are shown:

Example: Solve: 3x - 6 > 8x - 7

For example, to solve -3x is less than 12, divide both sides by -3, to get x is greater than -4.

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## Unit: Solving equations & inequalities

Linear equations with variables on both sides.

- Why we do the same thing to both sides: Variable on both sides (Opens a modal)
- Intro to equations with variables on both sides (Opens a modal)
- Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
- Equation with variables on both sides: fractions (Opens a modal)
- Equation with the variable in the denominator (Opens a modal)
- Equations with variables on both sides Get 3 of 4 questions to level up!
- Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

## Linear equations with parentheses

- Equations with parentheses (Opens a modal)
- Reasoning with linear equations (Opens a modal)
- Multi-step equations review (Opens a modal)
- Equations with parentheses Get 3 of 4 questions to level up!
- Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!
- Reasoning with linear equations Get 3 of 4 questions to level up!

## Analyzing the number of solutions to linear equations

- Number of solutions to equations (Opens a modal)
- Worked example: number of solutions to equations (Opens a modal)
- Creating an equation with no solutions (Opens a modal)
- Creating an equation with infinitely many solutions (Opens a modal)
- Number of solutions to equations Get 3 of 4 questions to level up!
- Number of solutions to equations challenge Get 3 of 4 questions to level up!

## Linear equations with unknown coefficients

- Linear equations with unknown coefficients (Opens a modal)
- Why is algebra important to learn? (Opens a modal)
- Linear equations with unknown coefficients Get 3 of 4 questions to level up!

## Multi-step inequalities

- Inequalities with variables on both sides (Opens a modal)
- Inequalities with variables on both sides (with parentheses) (Opens a modal)
- Multi-step inequalities (Opens a modal)
- Using inequalities to solve problems (Opens a modal)
- Multi-step linear inequalities Get 3 of 4 questions to level up!
- Using inequalities to solve problems Get 3 of 4 questions to level up!

## Compound inequalities

- Compound inequalities: OR (Opens a modal)
- Compound inequalities: AND (Opens a modal)
- A compound inequality with no solution (Opens a modal)
- Double inequalities (Opens a modal)
- Compound inequalities examples (Opens a modal)
- Compound inequalities review (Opens a modal)
- Solving equations & inequalities: FAQ (Opens a modal)
- Compound inequalities Get 3 of 4 questions to level up!

## About this unit

- Solve equations and inequalities
- Simplify expressions
- Factor polynomials
- Graph equations and inequalities
- Advanced solvers
- All solvers
- Arithmetics
- Determinant
- Percentages
- Scientific Notation
- Inequalities

## Equations and Inequalities Involving Signed Numbers

## 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

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

Dividing each side by -3, we obtain

## LITERAL EQUATIONS

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

## GRAPHING INEQUALITIES

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

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 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 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.

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

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

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

- 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.

## Math Topics

- Practice Problems
- Assignment Problems
- Show all Solutions/Steps/ etc.
- Hide all Solutions/Steps/ etc.
- Equations with Radicals
- Polynomial Inequalities
- Preliminaries
- Graphing and Functions
- Calculus II
- Calculus III
- Differential Equations
- Algebra & Trig Review
- Common Math Errors
- Complex Number Primer
- How To Study Math
- Cheat Sheets & Tables
- MathJax Help and Configuration
- Notes Downloads
- Complete Book
- Practice Problems Downloads
- Complete Book - Problems Only
- Complete Book - Solutions
- Assignment Problems Downloads
- Other Items
- Get URL's for Download Items
- Print Page in Current Form (Default)
- Show all Solutions/Steps and Print Page
- Hide all Solutions/Steps and Print Page

## Section 2.11 : Linear Inequalities

- \(4\left( {z + 2} \right) - 1 > 5 - 7\left( {4 - z} \right)\) Solution
- \(\displaystyle \frac{1}{2}\left( {3 + 4t} \right) \le 6\left( {\frac{1}{3} - \frac{1}{2}t} \right) - \frac{1}{4}\left( {2 + 10t} \right)\) Solution
- \( - 1 Solution
- \(8 \le 3 - 5z Solution
- \(0 \le 10w - 15 \le 23\) Solution
- \(\displaystyle 2 Solution
- If \(0 \le x Solution

## IMAGES

## VIDEO

## COMMENTS

The four steps for solving an equation include the combination of like terms, the isolation of terms containing variables, the isolation of the variable and the substitution of the answer into the original equation to check the answer.

The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...

Whether you love math or suffer through every single problem, there are plenty of resources to help you solve math equations. Skip the tutor and log on to load these awesome websites for a fantastic free equation solver or simply to find an...

Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own. · But these

Solving Inequalities · Add the same number to both sides. · Subtract the same number from both sides. · Multiply both sides by the same positive number. · Divide

Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a

It's like and equation, but with the inequality symbols, which are < and >. An equation uses an = (equal sign). For example: 3x + 8 = 2x - 4 is an equation.

There are lots of strategies we can use to solve equations. Let's explore some different ways to solve equations and inequalities.

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

Practice Page. Directions: Solve the following inequalities for the designated variable. The students' hints may be helpful.

Here is a set of practice problems to accompany the Linear Inequalities section of the Solving Equations and Inequalities chapter of the

This algebra video tutorial provides a basic introduction into how to solve linear inequalities. It explains how to graph the solution using

The expression 5x − 4 > 2x + 3 looks like an equation but with the equals sign replaced by an arrowhead. It is an example of an inequality. This denotes that

Key moments. View all · draw that on the number line · draw that on the number line · draw that on the number line · let me draw the number line.