Solving Linear Equations and Inequalities

Absolute Value Inequalities

Absolute value inequalities can be solved by writing two separate inequalities.
An inequality that contains an absolute value expression is called an absolute value inequality. Solving absolute value inequalities is similar to solving absolute value equations. Isolate the absolute value expression using the properties of inequality, if necessary. Then write a compound inequality, which is a set of two separate inequalities connected by and or or. For less-than inequalities, use and (LAND). For greater-than inequalities, use or (GORE). The type of compound inequality depends on the original inequality symbol.
Less-Than Inequalities
(<\lt, \leq)
Greater-Than Inequalities
(>\gt, \geq)
Description For Less-Than inequalities, use AND (LAND).

If the absolute value expression is Less than a number, write it as two separate inequalities using AND (LAND). This can also be written as a single inequality with the variable between the positive and negative numbers.
For Greater-than inequalities, use OR (GORE).

If the absolute value expression is Greater than a number, write it as two separate inequalities using OR (GORE). The solution set has two separate parts. So, it cannot be written as a single inequality.
Example x<7|x|\lt 7
x>7x\gt -7 AND x<7x\lt 7
7<x<7-7\lt x\lt 7
x7|x|\ge7
x7x\leq-7 OR x7x\geq7
Interval Notation (7,7)(-7, 7) (,7][7,)\left (-\infty,-7 \right ]\cup \left [7,\infty \right )

The symbol \cup means union and is used to describe the combination of two separate, or disjoint, sets.
Set-Builder Notation {x7<x<7}\{x|-7\lt x\lt 7\} {xx7 or x7}\{ x|x\leq-7 \text{ or } x\geq7\}

Examples of Inequality Number Lines
Less-than inequality: x<7|x|\lt 7
Solutions: x>7x\gt-7 and x<7x\lt7
Greater-than-or-equal-to inequality: x7|x|\ge7
Solutions: x7x \leq-7 or x7x\geq7

Step-By-Step Example
Solving Absolute Value Inequalities with Less Than
Solve the inequality:
x2+38\left|x-2\right|+3\leq8
Step 1
Isolate the absolute value expression. Apply the subtraction property of inequality to undo the addition. Subtract 3 from both sides.
x2+38x2+3383x25\begin{aligned}|x-2|+3 &\leq 8 \\ |x-2|+3 - 3 &\leq 8-3\\|x-2|&\leq5\end{aligned}
Step 2
Write the inequality as two separate inequalities using and.
x25andx25x-2\leq5\;\;\;\;\; \text{and} \;\;\;\;\;x-2\geq-5
Step 3
Solve both inequalities by applying the addition property of inequality to undo the subtraction. Add 2 to both sides.
x25x2+25+2x7andx25x2+25+2x3\begin{aligned}x-2&\leq5\\x-2+2&\leq5+2\\x&\leq7\end{aligned}\;\;\;\;\;\text{and}\;\;\;\;\;\begin{aligned}x-2&\geq-5\\x-2+2&\geq-5+2\\x&\geq-3 \end{aligned}
Solution
The two inequalities can be written as a single compound inequality:
3x7-3\leq x\leq7
It can also be written as [3,7]\left[-3,7\right] using interval notation and can be graphed on a number line.
Step-By-Step Example
Solving Absolute Value Inequalities with Greater Than
Solve the inequality:
2x+1>62|x+1|\gt6
Step 1
Isolate the absolute value expression. Apply the division property of inequality to undo the multiplication. Divide both sides by 2.
2x+1>62x+12>62x+1>3\begin{aligned}2\left|x+1\right|&\gt 6\\\frac{2\left|x+1\right|}{2}&\gt \frac{6}{2}\\\left|x+1\right|&\gt 3\end{aligned}
Step 2
Write the inequality as two separate inequalities, using or.
x+1>3orx+1<3x+1\gt 3\;\;\;\;\; \text{or} \;\;\;\;\;x+1\lt -3
Step 3
Solve both inequalities by applying the addition property of inequality to undo the subtraction. Subtract 1 from both sides.
x+1>3x+11>31x>2orx+1<3x+11<31x<4\begin{aligned}x+1&\gt 3 \\ x+1-1&\gt 3-1\\x&\gt 2\end{aligned} \;\;\;\;\; \text{or} \;\;\;\;\;\begin{aligned}x+1&\lt -3\\x+1-1&\lt -3-1\\x&\lt -4\end{aligned}
Solution
The inequalities x>2x\gt 2 or x<4x\lt -4 can be written in interval notation as (,4)(2,)\left (-\infty,-4 \right )\cup \left (2,\infty \right ) and can be graphed on a number line.