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.

Graphing Inequalities Example: ten larger or smaller

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.

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

Graph of Inequality: larger minus seven

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

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

Graph of Inequalitys: twenty or smaller

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

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

Graph of Inequality: greater than two

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

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

Graph of One-step Inequality: five or larger

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.

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

Graph of Two-step Inequality: larger minus four

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

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

Graph of Two-step Inequality: minus two or larger

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.

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

Graph of Multi-step Inequalities: smaller one one third

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

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

Graph of Multi-step Inequalities: smaller three

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.

Graph of AND-Inequality, Intersection: two or larger and five or smaller

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

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

Graph of OR-Inequality, Union: smaller two or larger five