Algebra I

Absolute Value Equations and Inequalities

Lesson

Absolute value measures distance from zero. So $|x| = 5$ asks: what numbers sit exactly 5 units from zero? There are two answers: $x = 5$ or $x = -5$ .

The rule

|A| = k \implies A = k \ \text{ or }\ A = -k

Whatever’s inside the bars can equal $+k$ or $-k$ . Solve both.

Worked example 1

|x - 3| = 5

Split into two equations:

x - 3 = 5 \quad \text{or} \quad x - 3 = -5

Solve each: $x = 8$ or $x = -2$ .

Worked example 2 — isolate the absolute value first

2|x| + 1 = 11

Subtract 1, then divide by 2:

|x| = 5 \implies x = 5 \ \text{ or }\ x = -5

Absolute value inequalities use the same idea, but the answer is a range.

Two patterns

“Less than”: $|A| < k$ means $-k < A < k$ — the inside is between the two boundary values (a sandwich).
“Greater than”: $|A| > k$ means $A < -k$ or $A > k$ — the inside is outside the two boundaries (two separate pieces).

No solution alert

If the absolute value equals a negative number (after isolating), there is no solution. Absolute value is never negative.

How to type your answer

Enter both solutions separated by a comma: 8,-2. Order doesn’t matter. If there’s only one solution, type just that number.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

|x| = 4

Problem 2

|x| = 9

Problem 3

|x| = 0

Problem 4

|x| + 1 = 6

Practice

Standard problems matching the lesson.

Problem 5

|x| = 12

Problem 6

|x - 3| = 5

Problem 7

|x + 4| = 6

Problem 8

|x - 2| = 7

Problem 9

|x + 1| = 3

Problem 10

2|x| = 10

Problem 11

3|x| = 12

Problem 12

|x| - 4 = 1

Problem 13

|x| + 2 = 9

Problem 14

|2x| = 8

Problem 15

|3x| = 9

Problem 16

|x - 5| = 0

Problem 17

Thermostat: |T - 70| = 3. Two boundary temperatures?

Problem 18

Measurement error |e| = 0.5. Two possible values?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

|x - 1| = 8

Problem 20

|2x - 3| = 5

Problem 21

|x + 6| - 2 = 7

Problem 22

2|x - 1| = 6

Problem 23

|2x + 4| = 10

Problem 24

Solve |x| < 6 — give the two boundary values.

Problem 25

Solve |x - 4| > 2 — give the two boundary values.

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