Absolute Value Equations Worksheet for 9th - 11th Grade | Lesson ... - Free Printable

Let’s solve each absolute value equation one by one. Remember:
Absolute value means distance from zero — so it’s always positive or zero.
If |x| = a, then x = a or x = -a (as long as a ≥ 0).
If the right side is negative, there’s no solution, because absolute value can’t be negative.

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We’ll go problem by problem:

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(1) |3x + 4| = 42

Set up two equations:
- 3x + 4 = 42 → 3x = 38 → x = 38/3
- 3x + 4 = -42 → 3x = -46 → x = -46/3

✔ Solutions: x = 38/3, x = -46/3

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(2) |-4x| = 20

This is same as |4x| = 20 (since |-4x| = |4x|)

So:
- 4x = 20 → x = 5
- 4x = -20 → x = -5

✔ Solutions: x = 5, x = -5

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(3) |x - 10| + 7 = 7

Subtract 7 from both sides:
|x - 10| = 0

Only one solution when absolute value equals 0:
x - 10 = 0 → x = 10

✔ Solution: x = 10

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(4) |(1/2)x| = 3

Multiply both sides by 2 to eliminate fraction? Or just solve directly:

(1/2)x = 3 → x = 6
(1/2)x = -3 → x = -6

✔ Solutions: x = 6, x = -6

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(5) |7 + x| = -1

Wait! Absolute value cannot equal a negative number.

✘ No solution

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(6) |-2x| = 19

Same as |2x| = 19

So:
2x = 19 → x = 19/2
2x = -19 → x = -19/2

✔ Solutions: x = 19/2, x = -19/2

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(7) 5|x| = 30

Divide both sides by 5:
|x| = 6

So:
x = 6 or x = -6

✔ Solutions: x = 6, x = -6

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(8) |(1/4)x| = 3

Multiply both sides by 4:
|x| = 12

So:
x = 12 or x = -12

✔ Solutions: x = 12, x = -12

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(9) -5|x| = -12

Divide both sides by -5:
|x| = 12/5

So:
x = 12/5 or x = -12/5

✔ Solutions: x = 12/5, x = -12/5

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(10) |4x| = 11

So:
4x = 11 → x = 11/4
4x = -11 → x = -11/4

✔ Solutions: x = 11/4, x = -11/4

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(11) |x| - 3 = 2

Add 3 to both sides:
|x| = 5

So:
x = 5 or x = -5

✔ Solutions: x = 5, x = -5

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(12) |(x/2)| = 5

Multiply both sides by 2:
|x| = 10

So:
x = 10 or x = -10

✔ Solutions: x = 10, x = -10

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(13) |(2(x+1))/3| = 5

First, multiply both sides by 3:
|2(x+1)| = 15

Now divide both sides by 2? Or keep inside:

2(x+1) = 15 → x+1 = 15/2 → x = 15/2 - 1 = 13/2
OR
2(x+1) = -15 → x+1 = -15/2 → x = -15/2 - 1 = -17/2

✔ Solutions: x = 13/2, x = -17/2

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(14) 4|x + 4| = 56

Divide both sides by 4:
|x + 4| = 14

So:
x + 4 = 14 → x = 10
x + 4 = -14 → x = -18

✔ Solutions: x = 10, x = -18

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(15) |7x| - 3 = 73

Add 3 to both sides:
|7x| = 76

So:
7x = 76 → x = 76/7
7x = -76 → x = -76/7

✔ Solutions: x = 76/7, x = -76/7

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(16) |(1/2)x - 4| = 7

Set up two cases:

Case 1: (1/2)x - 4 = 7 → (1/2)x = 11 → x = 22
Case 2: (1/2)x - 4 = -7 → (1/2)x = -3 → x = -6

✔ Solutions: x = 22, x = -6

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(17) |(x+4)/4| = 3

Multiply both sides by 4:
|x + 4| = 12

So:
x + 4 = 12 → x = 8
x + 4 = -12 → x = -16

✔ Solutions: x = 8, x = -16

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(18) -6|6x + 2| = -78

Divide both sides by -6:
|6x + 2| = 13

Now:
6x + 2 = 13 → 6x = 11 → x = 11/6
6x + 2 = -13 → 6x = -15 → x = -15/6 = -5/2

✔ Solutions: x = 11/6, x = -5/2

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Final Answer:
(1) x = 38/3, x = -46/3
(2) x = 5, x = -5
(3) x = 10
(4) x = 6, x = -6
(5) No solution
(6) x = 19/2, x = -19/2
(7) x = 6, x = -6
(8) x = 12, x = -12
(9) x = 12/5, x = -12/5
(10) x = 11/4, x = -11/4
(11) x = 5, x = -5
(12) x = 10, x = -10
(13) x = 13/2, x = -17/2
(14) x = 10, x = -18
(15) x = 76/7, x = -76/7
(16) x = 22, x = -6
(17) x = 8, x = -16
(18) x = 11/6, x = -5/2