## Introduction

**Inequalities** are similar to **equations**, but their solution sets include values that are **less than**, **less than or equal to**, **greater than** and **greater than or equal to**.

These inequalities model the **four different inequality symbols**:

- $x<10$
- $x\leq10$
- $x>10$
- $x\geq10$

## Graphing Inequalities

To graph the solution set of an inequality, pay close attention to the symbol used in the problem. An **open circle** indicates **less than or greater than**, whereas a

**closed circle**indicates

**less than or equal to**.

*or*greater than or equal to## One-Step Inequalities

To solve **one-step inequalities**, follow the same procedures as when solving one-step equations with one exception. **When multiplying or dividing by a negative number, remember to flip the inequality symbol.**

To solve this one-step inequality, you can use mental math.

$\begin{array}{lcr} -2x&\lt &14 \\ \frac{-2}{-2}x&\lt& \frac{14}{-2} \\ ~~~~~x&\gt& -7x \end{array}$

Use the **opposite operation** to undo the constant found in this inequality.

$\begin{array}{lcr} x -5&\leq&15 \\ x -5 +5&\leq&15 +5 \\ x&\leq&20 \end{array}$

When solving one-step inequalities, watch out for graphing errors. The graph of this problem has an open circle.

$\begin{array}{lcr} x +3 &\gt& 5 \\ x+ 3 -3&\gt&5 -3 \\ x&\gt&2 \end{array}$

The graph of this one-step inequality has a closed circle.

$\begin{array}{lcr} ~2x&\geq&10 \\ \frac{2}{2}x&\geq&\frac{10}{2} \\ ~~~x&\geq&5 \end{array}$

## Two-Step Inequalities

Just like one-step inequalities, **two-step inequalities** follow the same procedures as equations, but watch out for multiplying or dividing by negative numbers.

To solve this problem, use two steps, but watch out for the negative coefficient.

$\begin{array}{lcr} ~~~7&\gt&-2x -1 \\ ~~~7 +1&\gt&-2x -1 +1 \\ ~~~8&\gt&-2x \\ ~\frac{8}{-2}&\gt&\frac{-2}{-2}x \\ -4&\lt& x \end{array}$

To solve this two-step inequality, use the **multiplicative inverse to undo the coefficient**.

$\begin{array}{lcr} \frac{x}{2} -8&\geq& -9 \\ \frac{x}{2} -8 +8&\geq& -9 +8 \\ \frac{x}{2}&\geq& -1 \\ \frac{x}{2}\times2&\geq& -1\times2 \\ x&\geq& -2 \end{array}$

## Multi-Step Inequalities

Similar to multi-step equations, **multi-step inequalities** can have numbers and variables on either side of the **inequality sign**. **Combine like terms first**, and remember to watch out for sign changes due to multiplying or dividing by a negative number.

The graph of this multi-step inequality has a **pen dot**.

$\begin{array}{lcr} ~8x +4 -2x&\lt&12 \\ ~6x +4&\lt&12 \\ ~6x +4 -4&\lt&12 -4 \\ ~6x&\lt&8 \\ \frac{6}{6}x&\lt&\frac{8}{6}\\ ~~~x&\lt&\frac{8}{6} \\ ~~~x&\lt&1\frac{1}{3} \end{array}$

When solving inequalities with **negative numbers**, it’s smart to **check if the inequality changes direction**.

$\begin{array}{lcr} -18&\leq& -5x + -x \\ -18&\leq& -6x \\ ~\frac{-18}{-6}&\leq&\frac{-6}{-6}x \\ ~~~3&\geq& x \end{array}$

## Compound Inequalities

**Compound inequalities** are also known as *and* and *or* compound inequalities because they contain inequalities that are separated by the words *and* and *or*.

This inequality can be written as two inequalities separated by the word and.

$\begin{array}{lcccr} 2~\leq~~~&x&~~~\leq~5 \\ x~\geq~2& \text{and}&x~\leq~5 \end{array}$

**Graphs of AND inequalities** show **only one piece of the graph shaded**. The *and* represents an **intersection**.

$x\leq2 \quad \text{or} \quad x\geq5$

**Graphs of OR inequalities** show **two separate pieces of the graph shaded**. The *or* represents a **union**.