Solving Absolute Value Equations and Inequalities WS | PDF ... - Free Printable

Let’s solve each absolute value equation step by step, and check each solution to ensure it satisfies the original equation.

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a. |x – 6| = 8

Step 1: Absolute value equations split into two cases:
> Case 1: x – 6 = 8
> Case 2: x – 6 = –8

Solve Case 1:
x – 6 = 8 → x = 8 + 6 → x = 14

Solve Case 2:
x – 6 = –8 → x = –8 + 6 → x = –2

✔ Solutions: x = 14 or x = –2

Check:
- |14 – 6| = |8| = 8 ✔️
- |–2 – 6| = |–8| = 8 ✔️

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b. |x + 2| = –8

Step 1: The absolute value of any real number is always ≥ 0. It can never be negative.

So |x + 2| = –8 has no solution.

✘ No solution

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c. |–9x| = 64

Note: |–9x| = |–9|·|x| = 9|x|, so this is equivalent to 9|x| = 64 → |x| = 64/9

But we can also solve directly:

Step 1: Split into two cases:
> Case 1: –9x = 64
> Case 2: –9x = –64

Case 1:
–9x = 64 → x = 64 / (–9) → x = –64/9

Case 2:
–9x = –64 → x = (–64)/(–9) → x = 64/9

✔ Solutions: x = 64/9 or x = –64/9

Check:
- |–9*(64/9)| = |–64| = 64 ✔️
- |–9*(–64/9)| = |64| = 64 ✔️

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d. |–7x + 4| = 18

Step 1: Split into two cases:
> Case 1: –7x + 4 = 18
> Case 2: –7x + 4 = –18

Case 1:
–7x + 4 = 18
→ –7x = 14
→ x = 14 / (–7) → x = –2

Case 2:
–7x + 4 = –18
→ –7x = –22
→ x = (–22)/(–7) → x = 22/7

✔ Solutions: x = –2 or x = 22/7

Check:
- |–7(–2) + 4| = |14 + 4| = |18| = 18 ✔️
- |–7*(22/7) + 4| = |–22 + 4| = |–18| = 18 ✔️

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e. |4x + 4| = 28

Step 1: Split into two cases:
> Case 1: 4x + 4 = 28
> Case 2: 4x + 4 = –28

Case 1:
4x + 4 = 28 → 4x = 24 → x = 6

Case 2:
4x + 4 = –28 → 4x = –32 → x = –8

✔ Solutions: x = 6 or x = –8

Check:
- |4(6) + 4| = |24 + 4| = |28| = 28 ✔️
- |4(–8) + 4| = |–32 + 4| = |–28| = 28 ✔️

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f. 5|n + 10| = 10

Step 1: Divide both sides by 5 first:
|n + 10| = 2

Now split into two cases:
> Case 1: n + 10 = 2
> Case 2: n + 10 = –2

Case 1:
n + 10 = 2 → n = –8

Case 2:
n + 10 = –2 → n = –12

✔ Solutions: n = –8 or n = –12

Check:
- 5|–8 + 10| = 5|2| = 5×2 = 10 ✔️
- 5|–12 + 10| = 5|–2| = 5×2 = 10 ✔️

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g. |–6n| + 3 = 46

Step 1: Subtract 3 from both sides:
|–6n| = 43

Since |–6n| = 6|n|, we have 6|n| = 43 → |n| = 43/6

But let’s solve directly:

Split into two cases:
> Case 1: –6n = 43
> Case 2: –6n = –43

Case 1:
–6n = 43 → n = –43/6

Case 2:
–6n = –43 → n = 43/6

✔ Solutions: n = 43/6 or n = –43/6

Check:
- |–6*(43/6)| + 3 = |–43| + 3 = 43 + 3 = 46 ✔️
- |–6*(–43/6)| + 3 = |43| + 3 = 43 + 3 = 46 ✔️

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h. |3v – 2| / 5 = 4

Step 1: Multiply both sides by 5:
|3v – 2| = 20

Split into two cases:
> Case 1: 3v – 2 = 20
> Case 2: 3v – 2 = –20

Case 1:
3v – 2 = 20 → 3v = 22 → v = 22/3

Case 2:
3v – 2 = –20 → 3v = –18 → v = –6

✔ Solutions: v = 22/3 or v = –6

Check:
- |3*(22/3) – 2| / 5 = |22 – 2| / 5 = |20| / 5 = 20/5 = 4 ✔️
- |3*(–6) – 2| / 5 = |–18 – 2| / 5 = |–20| / 5 = 20/5 = 4 ✔️

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## ✔ Final Answers:

a. x = 14 or x = –2
b. No solution
c. x = 64/9 or x = –64/9
d. x = –2 or x = 22/7
e. x = 6 or x = –8
f. n = –8 or n = –12
g. n = 43/6 or n = –43/6
h. v = 22/3 or v = –6

All solutions checked and verified!