How to Solve Absolute Value Inequalities

Q: What does |x − a| < b mean on a number line?

The numbers whose distance from a is less than b. They fill the stretch from a − b to a + b, centered at a.

Q: Do I still flip the sign when I divide by a negative?

Yes. Isolating −2|x| ≤ −6 means dividing by −2, which gives |x| ≥ 3.

Quick answer

An absolute value inequality is a question about distance. |x − 3| < 5 asks for the numbers within 5 of 3, which form one stretch of the number line: −2 < x < 8. |x − 3| > 5 asks for the numbers more than 5 away from 3, which form two rays: x < −2 or x > 8. Isolate the absolute value first, then write the matching AND or OR compound inequality and solve it. A negative number on the right side makes the answer every number or no number at all.

What you'll learn

Turn an absolute value inequality into an AND or an OR compound inequality
Explain both forms with distance on the number line
Solve and graph absolute value inequalities, including ≤ and ≥
Recognize inequalities that hold for every number or for none
Write an absolute value inequality for a tolerance

Distance, again

The absolute value $|x - 3|$ is the distance from $x$ to $3$ on the number line. So the inequality

|x - 3| < 5

asks a distance question: which numbers are less than $5$ away from $3$ ? Walk $5$ units each way from $3$ and you reach $-2$ and $8$ . Every number strictly between them is closer than that:

|x − 3| < 5: every number within 5 of 3

The solution is $-2 < x < 8$ . Now flip the question. $|x - 3| > 5$ asks which numbers are more than $5$ away from $3$ . Those lie past $-2$ on the left or past $8$ on the right:

|x − 3| > 5: every number farther than 5 from 3

The solution is $x < -2$ or $x > 8$ .

The two forms

For a positive number $c$ and any expression $u$ :

Inequality	Meaning	Compound form
$\lvert u \rvert < c$	$u$ is within $c$ of zero	$-c < u < c$
$\lvert u \rvert > c$	$u$ is farther than $c$ from zero	$u < -c$ or $u > c$

The same forms hold for $\le$ and $\ge$ . The endpoints are then included.

Why less than means AND

To be within $5$ of zero, a number must pass two tests at once. It must not be too far right, so $u < 5$ . It must not be too far left, so $u > -5$ . Both conditions hold together, which is an AND. To be farther than $5$ from zero, a number needs only one escape: far right or far left. Less than traps the value between two walls, so both conditions must hold; greater than lets it escape in either direction, so one condition is enough.

Worked examples

Common mistakes

Practice problems

Solve $|x| < 6$ .

Answer

$-6 < x < 6$

Full solution

The numbers within $6$ of zero lie between $-6$ and $6$ .
Solve $|x - 1| \ge 4$ .

Answer

$x \le -3$ or $x \ge 5$

Full solution

Greater than or equal gives an OR: $x - 1 \le -4$ or $x - 1 \ge 4$ , so $x \le -3$ or $x \ge 5$ .
Solve $|x + 2| \le 5$ .

Answer

$-7 \le x \le 3$

Full solution

$-5 \le x + 2 \le 5$ . Subtract $2$ from all three parts.
Solve $|2x + 3| < 9$ .

Answer

$-6 < x < 3$

Full solution

$-9 < 2x + 3 < 9$ , so $-12 < 2x < 6$ and $-6 < x < 3$ .
Solve $2|x - 4| - 3 \le 7$ .

Hint

Isolate the absolute value before you split.

Answer

$-1 \le x \le 9$

Full solution

Add $3$ and divide by $2$ : $|x - 4| \le 5$ . Then $-5 \le x - 4 \le 5$ , so $-1 \le x \le 9$ .
Solve $|5 - x| > 1$ .

Answer

$x < 4$ or $x > 6$

Full solution

$5 - x < -1$ gives $-x < -6$ , so $x > 6$ . And $5 - x > 1$ gives $-x > -4$ , so $x < 4$ . Each step divides by $-1$ and flips the sign.
Solve $|x + 7| < -2$ , and then $|x + 7| > -2$ .

Answer

No solution; every real number

Full solution

A distance is never negative. It is never less than $-2$ , and it is always greater than $-2$ .
Write an absolute value inequality whose solution is $1 \le x \le 9$ .

Hint

Find the center of the stretch and its distance to each end.

Answer

$|x - 5| \le 4$

Full solution

The center is $\tfrac{1 + 9}{2} = 5$ , and each end is $4$ units from it. So the solution is every number within $4$ of $5$ .
A thermostat keeps a room within $3$ degrees of $68$ °F. Write and solve an inequality for the temperature $T$ .

Answer

$|T - 68| \le 3$ , so $65 \le T \le 71$

Full solution

Within $3$ of $68$ means $-3 \le T - 68 \le 3$ . Add $68$ to all three parts.
A student solves $|x - 2| > 6$ and writes $-4 < x < 8$ . What went wrong?

Hint

Test $x = 0$ in the original inequality.

Answer

That is the solution of $|x - 2| < 6$ . The correct answer is $x < -4$ or $x > 8$ .

Full solution

At $x = 0$ , $|0 - 2| = 2$ , which is not greater than $6$ , yet $0$ is in the student’s answer. Greater than means farther than $6$ from $2$ , which gives two rays: $x - 2 < -6$ or $x - 2 > 6$ .

Frequently asked questions

How do you solve an absolute value inequality?

Isolate the absolute value. If it is less than a positive number c, write −c < inside < c and solve. If it is greater than c, write inside < −c or inside > c and solve both parts.

Why does less than give AND, and greater than give OR?

|x| < 5 means within 5 of zero: one stretch between −5 and 5, where two conditions hold at once. |x| > 5 means farther than 5 from zero, which can happen in either of two directions.

What if the number on the right side is negative?

A distance is never negative. So |x − 1| < −2 has no solution, and |x − 1| > −2 is true for every number.

What does |x − a| < b mean on a number line?