Absolute value inequalities

Introduction

An inequality that contains one or more absolute values is called an absolute value inequality. For example, the following are absolute value inequalities:

$4|y|\le 8$,   $|5x-3|\gt 4$,   $|2u-8|-5\lt 10$

To solve an absolute value inequality, first isolate the absolute value on one side, then apply one of the four formulas below.

The four formulas

In the formulas below, $X$ stands for the expression inside the absolute value, $p$ stands for a positive number, and $n$ stands for a negative number.

Formula 1. If $|X|\lt p$, then $-p\lt X\lt p$. If $|X|\le p$, then $-p\le X\le p$.

For example, $|x|\le 4$ is the set of all numbers whose absolute value is less than or equal to $4$:

$-4\le x\le 4$

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Formula 2. If $|X|\lt n$ or $|X|\le n$, then the inequality has no solution, because an absolute value can never be less than a negative number.

For example, $|x|\le -5$ has no solution.

Formula 3. If $|X|\gt p$, then $X\lt -p$ or $X\gt p$. If $|X|\ge p$, then $X\le -p$ or $X\ge p$.

For example, $|x|\ge 4$ is the set of all numbers whose absolute value is greater than or equal to $4$:

$x\le -4$   or   $x\ge 4$

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Formula 4. If $|X|\gt n$ or $|X|\ge n$, then all real numbers are solutions, because the absolute value of any real number is greater than a negative number.

For example, $|x|\ge -4$ is true for all real numbers.

Worked examples

Example 1: Solve $|4x+2|\lt 10$, and graph the solution on the number line.

Solution: The absolute value is already isolated. This has the form $|X|\lt p$ with $p=10$, so use Formula 1:

$-10\lt 4x+2\lt 10$

Subtract $2$ from all three parts:

$-12\lt 4x\lt 8$

Divide all three parts by $4$:

$-3\lt x\lt 2$

Solution in set-builder notation: $\{x\mid -3\lt x\lt 2\}$.

Solution in interval notation: $(-3,2)$.

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Example 2: Solve $|5w+1|-4\le 7$.

Solution: First isolate the absolute value by adding $4$ to both sides:

$|5w+1|\le 11$

This has the form $|X|\le p$ with $p=11$, so use Formula 1:

$-11\le 5w+1\le 11$

Subtract $1$ from all three parts:

$-12\le 5w\le 10$

Divide all three parts by $5$:

$-\dfrac{12}{5}\le w\le 2$

Solution in set-builder notation: $\left\{w\mid -\dfrac{12}{5}\le w\le 2\right\}$.

Solution in interval notation: $\left[-\dfrac{12}{5},\,2\right]$.

Example 3: Solve $2|y+1|+4\lt -18$.

Solution: First isolate the absolute value. Subtract $4$ from both sides, then divide by $2$:

$\begin{align*}2|y+1|&\lt -22\\|y+1|&\lt -11\end{align*}$

The absolute value is less than a negative number, so by Formula 2 the inequality has no solution.

Solution: $\varnothing$ (the empty set).

Example 4: Solve $|w-6|\ge 8$, and graph the solution on the number line.

Solution: The absolute value is already isolated. This has the form $|X|\ge p$ with $p=8$, so use Formula 3:

$w-6\le -8$   or   $w-6\ge 8$

Add $6$ to both sides of each inequality:

$w\le -2$   or   $w\ge 14$

Solution in set-builder notation: $\{w\mid w\le -2 \text{ or } w\ge 14\}$.

Solution in interval notation: $(-\infty,-2]\cup[14,\infty)$.

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Example 5: Solve $|2x-21|+20\gt 7$.

Solution: First isolate the absolute value by subtracting $20$ from both sides:

$|2x-21|\gt -13$

The absolute value is greater than a negative number, so by Formula 4 every real number is a solution.

Solution in set-builder notation: $\{x\mid x \text{ is any real number}\}$.

Solution in interval notation: $(-\infty,\infty)$.

Example 6: Solve $4|x+5|-11\gt 9$.

Solution: First isolate the absolute value. Add $11$ to both sides, then divide by $4$:

$\begin{align*}4|x+5|&\gt 20\\|x+5|&\gt 5\end{align*}$

This has the form $|X|\gt p$ with $p=5$, so use Formula 3:

$x+5\lt -5$   or   $x+5\gt 5$

Subtract $5$ from both sides of each inequality:

$x\lt -10$   or   $x\gt 0$

Solution in set-builder notation: $\{x\mid x\lt -10 \text{ or } x\gt 0\}$.

Solution in interval notation: $(-\infty,-10)\cup(0,\infty)$.

Example 7: Solve $|1-4x|+6\lt 11$.

Solution: First isolate the absolute value by subtracting $6$ from both sides:

$|1-4x|\lt 5$

This has the form $|X|\lt p$ with $p=5$, so use Formula 1:

$-5\lt 1-4x\lt 5$

Subtract $1$ from all three parts:

$-6\lt -4x\lt 4$

Divide all three parts by $-4$. Dividing by a negative number reverses the inequalities:

$\dfrac{3}{2}\gt x\gt -1$

Writing it with the smaller number first, $-1\lt x\lt \dfrac{3}{2}$.

Solution in set-builder notation: $\left\{x\mid -1\lt x\lt \dfrac{3}{2}\right\}$.

Solution in interval notation: $\left(-1,\,\dfrac{3}{2}\right)$.