To solve these absolute value equations, we use the property that if $|A| = B$, then $A = B$ or $A = -B$. We'll apply this to each equation step by step.
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1) |a - 4| = 8
→ a - 4 = 8 → a = 12
→ a - 4 = -8 → a = -4
✔ a = 12 or -4
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2) |6 + b| = 10
→ 6 + b = 10 → b = 4
→ 6 + b = -10 → b = -16
✔ b = 4 or -16
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3) |9 - c| = 4
→ 9 - c = 4 → -c = -5 → c = 5
→ 9 - c = -4 → -c = -13 → c = 13
✔ c = 5 or 13
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4) |3 + d + 5| = 7 → |d + 8| = 7
→ d + 8 = 7 → d = -1
→ d + 8 = -7 → d = -15
✔ d = -1 or -15
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5) |2e| = 24
→ 2e = 24 → e = 12
→ 2e = -24 → e = -12
✔ e = 12 or -12
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6) |2f + 1| = 13
→ 2f + 1 = 13 → 2f = 12 → f = 6
→ 2f + 1 = -13 → 2f = -14 → f = -7
✔ f = 6 or -7
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7) 3|g - 4| = 12 → |g - 4| = 4
→ g - 4 = 4 → g = 8
→ g - 4 = -4 → g = 0
✔ g = 8 or 0
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8) ½|h + 2| = 3 → |h + 2| = 6
→ h + 2 = 6 → h = 4
→ h + 2 = -6 → h = -8
✔ h = 4 or -8
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9) |4i| - |-7| = 1 → |4i| - 7 = 1 → |4i| = 8
→ 4i = 8 → i = 2
→ 4i = -8 → i = -2
✔ i = 2 or -2
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10) |2(j - 3)| = 6 → 2|j - 3| = 6 → |j - 3| = 3
→ j - 3 = 3 → j = 6
→ j - 3 = -3 → j = 0
✔ j = 6 or 0
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11) |4(k + 3)| = 20 → 4|k + 3| = 20 → |k + 3| = 5
→ k + 3 = 5 → k = 2
→ k + 3 = -5 → k = -8
✔ k = 2 or -8
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12) |8 - 2l| = 6
→ 8 - 2l = 6 → -2l = -2 → l = 1
→ 8 - 2l = -6 → -2l = -14 → l = 7
✔ l = 1 or 7
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13) 3|m - 5| = 21 → |m - 5| = 7
→ m - 5 = 7 → m = 12
→ m - 5 = -7 → m = -2
✔ m = 12 or -2
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14) |2n - 3| = 11
→ 2n - 3 = 11 → 2n = 14 → n = 7
→ 2n - 3 = -11 → 2n = -8 → n = -4
✔ n = 7 or -4
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15) |10 - 2o| = 4
→ 10 - 2o = 4 → -2o = -6 → o = 3
→ 10 - 2o = -4 → -2o = -14 → o = 7
✔ o = 3 or 7
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16) |3(p - 2)| = 24 → 3|p - 2| = 24 → |p - 2| = 8
→ p - 2 = 8 → p = 10
→ p - 2 = -8 → p = -6
✔ p = 10 or -6
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17) 5|4 - q| = 30 → |4 - q| = 6
→ 4 - q = 6 → -q = 2 → q = -2
→ 4 - q = -6 → -q = -10 → q = 10
✔ q = -2 or 10
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18) ½|r + 6| = 12 → |r + 6| = 24
→ r + 6 = 24 → r = 18
→ r + 6 = -24 → r = -30
✔ r = 18 or -30
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19) |½(s - 3)| = 7 → ½|s - 3| = 7 → |s - 3| = 14
→ s - 3 = 14 → s = 17
→ s - 3 = -14 → s = -11
✔ s = 17 or -11
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20) |6 - ¼t| = 4
→ 6 - ¼t = 4 → -¼t = -2 → t = 8
→ 6 - ¼t = -4 → -¼t = -10 → t = 40
✔ t = 8 or 40
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Final Answer:
1) 12 or -4
2) 4 or -16
3) 5 or 13
4) -1 or -15
5) 12 or -12
6) 6 or -7
7) 8 or 0
8) 4 or -8
9) 2 or -2
10) 6 or 0
11) 2 or -8
12) 1 or 7
13) 12 or -2
14) 7 or -4
15) 3 or 7
16) 10 or -6
17) -2 or 10
18) 18 or -30
19) 17 or -11
20) 8 or 40
Parent Tip: Review the logic above to help your child master the concept of solving absolute value equations worksheet.
