## 0.6 Inequalities

### QUICK REFERENCE

Inequality Signs

Symbol Words Example
$\neq$ not equal to ${2}\neq{8}$, 2 is not equal to 8.
$\gt$ greater than ${5}\gt{1}$, 5 is greater than 1
$\lt$ less than ${2}\lt{11}$, 2 is less than 11
$\geq$ greater than or equal to ${4}\geq{ 4}$, 4 is greater than or equal to 4
$\leq$ less than or equal to ${7}\leq{9}$, 7 is less than or equal to 9

Addition and Subtraction Properties of Inequalities

If $a>b$, then $a+c>b+c$.

If $a>b$, then $a−c>b−c$.

Multiplication and Division Properties of Inequality

 Start With Multiply By Final Inequality $a>b$ $c$ $ac>bc$ $a>b$ $-c$ $ac  Start With Divide By Final Inequality [latex]a>b$ $c$ $\displaystyle \frac{a}{c}>\frac{b}{c}$ $a>b$ $-c$ $\displaystyle \frac{a}{c}<\frac{b}{c}$

## Inequalities

First, let’s define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right—just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.

• ${x}\lt{9}$ indicates the list of numbers that are less than 9. Would you rather write ${x}\lt{9}$ or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)
• $-5\le{t}$ indicates all the numbers that are greater than or equal to $-5$.

Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.

For example:

• $x\lt5$ means all the real numbers that are less than 5, whereas;
• $5\lt{x}$ means that 5 is less than x, or we could rewrite this with the x on the left: $x\gt{5}$ note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.

### Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it’s easy to get tangled up in inequalities, just remember to read them from left to right.

Symbol Words Example
$\neq$ not equal to ${2}\neq{8}$, 2 is not equal to 8.
$\gt$ greater than ${5}\gt{1}$, 5 is greater than 1
$\lt$ less than ${2}\lt{11}$, 2 is less than 11
$\geq$ greater than or equal to ${4}\geq{ 4}$, 4 is greater than or equal to 4
$\leq$ less than or equal to ${7}\leq{9}$, 7 is less than or equal to 9

The inequality $x>y$ can also be written as ${y}<{x}$. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.

## Solve Single-Step Inequalities

### Solve inequalities with addition and subtraction

You can solve most inequalities using inverse operations as you did for solving equations.  This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.

### Addition and Subtraction Properties of Inequality

If $a>b$, then $a+c>b+c$.

If $a>b$, then $a−c>b−c$.

### Example

Solve for x.

${x}+3\lt{5}$

Just as you can check the solution to an equation, you can check a solution to an inequality. First, you check the endpoint by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions.

The example below shows how you could check that $x<2$ is the solution to $x+3<5$.

### Example

Check that $x<2$ is the solution to $x+3<5$.

Substitute the end point 2 into the related equation, $x+3=5$.

$\begin{array}{r}x+3=5 \\ 2+3=5 \\ 5=5\end{array}$

Pick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)

$\displaystyle \begin{array}{r}x+3<5 \\ 0+3<5 \\ 3<5\end{array}$

It checks!

$x<2$ is the solution to $x+3<5$.

The following examples show inequality problems that include operations with negative numbers.

### Example

Solve for x: $x-10\leq-12$

Check the solution to $x-10\leq -12$

### Example

Solve for a. $a-17>-17$

Check the solution to $a-17>-17$

What would you do if the variable were on the right side of the inequality?  In the following example, you will see how to handle this scenario.

### Example

Solve for x: $4\geq{x}+5$

Check the solution to $4\geq{x}+5$

### Solve inequalities with multiplication and division

Solving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let’s look at what happens to the inequality when you multiply or divide each side by the same number.

 Let’s start with the true statement: $10>5$ Let’s try again by starting with the same true statement: $10>5$ Next, multiply both sides by the same positive number: $10\cdot 2>5\cdot 2$ This time, multiply both sides by the same negative number: $10\cdot-2>5 \\ \,\,\,\,\,\cdot -2\,\cdot-2$ 20 is greater than 10, so you still have a true inequality: $20>10$ Wait a minute! $−20$ is not greater than $−10$, so you have an untrue statement. $−20>−10$ When you multiply by a positive number, leave the inequality sign as it is! You must “reverse” the inequality sign to make the statement true: $−20<−10$

Caution!  When you multiply or divide by a negative number, “reverse” the inequality sign.   Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.

### Multiplication and Division Properties of Inequality

 Start With Multiply By Final Inequality $a>b$ $c$ $ac>bc$ $a>b$ $-c$ $ac  Start With Divide By Final Inequality [latex]a>b$ $c$ $\displaystyle \frac{a}{c}>\frac{b}{c}$ $a>b$ $-c$ $\displaystyle \frac{a}{c}<\frac{b}{c}$

Keep in mind that you only change the sign when you are multiplying and dividing by a negative number. If you add or subtract by a negative number, the inequality stays the same.

### Example

Solve for x. $3x>12$

There was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive 3. In the next example, there is division by a negative number, so there is an additional step in the solution.

### Example

Solve for x. $−2x>6$

## Combine properties of inequality to solve algebraic inequalities

A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other.

### Example

Solve for p. $4p+5<29$

Check the solution.

### Example

Solve for x:  $3x–7\ge 41$

Check the solution.

When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.

### Example

Solve for p. $−58>14−6p$

Check the solution.

## Simplify and solve algebraic inequalities using the distributive property

As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.

### Example

Solve for x. $2\left(3x–5\right)\leq 4x+6$

Check the solution.