Let’s solve each inequality one by one. I’ll go step by step so you can follow along easily.
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1. 6(n + 10) ≥ 31
First, divide both sides by 6:
→ n + 10 ≥ 31/6
→ n + 10 ≥ 5.166... (or keep as fraction: 31/6)
Now subtract 10 from both sides:
→ n ≥ 31/6 - 10
→ n ≥ 31/6 - 60/6
→ n ≥ -29/6
→ n ≥ -4.833...
✔ Final Answer for #1:
n ≥ -29/6 or
n ≥ -4.83 (rounded)
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2. 6(y + 6) ≥ 35
Divide both sides by 6:
→ y + 6 ≥ 35/6
Subtract 6:
→ y ≥ 35/6 - 36/6
→ y ≥ -1/6
✔ Final Answer for #2:
y ≥ -1/6
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3. -8 ≤ m/10 - 10
Add 10 to both sides:
→ 2 ≤ m/10
Multiply both sides by 10:
→ 20 ≤ m
→ m ≥ 20
✔ Final Answer for #3:
m ≥ 20
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4. 5 ≤ y/(-3) + 19
Subtract 19 from both sides:
→ 5 - 19 ≤ y/(-3)
→ -14 ≤ y/(-3)
Multiply both sides by -3 → remember to flip the inequality sign!
→ (-14)(-3) ≥ y
→ 42 ≥ y
→ y ≤ 42
✔ Final Answer for #4:
y ≤ 42
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5. 6r + 10 ≥ 19
Subtract 10:
→ 6r ≥ 9
Divide by 6:
→ r ≥ 9/6 = 3/2 = 1.5
✔ Final Answer for #5:
r ≥ 1.5
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6. 7g + 18 < 4g + 10
Subtract 4g from both sides:
→ 3g + 18 < 10
Subtract 18:
→ 3g < -8
Divide by 3:
→ g < -8/3 ≈ -2.666...
✔ Final Answer for #6:
g < -8/3
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7. 5m + 2 < 9m + 1
Subtract 5m:
→ 2 < 4m + 1
Subtract 1:
→ 1 < 4m
Divide by 4:
→ 1/4 < m
→ m > 0.25
✔ Final Answer for #7:
m > 1/4
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8. -21 > -2m + 7
Subtract 7:
→ -28 > -2m
Divide by -2 → flip inequality!
→ 14 < m
→ m > 14
✔ Final Answer for #8:
m > 14
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9. 58 ≥ 14 + 11w
Subtract 14:
→ 44 ≥ 11w
Divide by 11:
→ 4 ≥ w
→ w ≤ 4
✔ Final Answer for #9:
w ≤ 4
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10. -5(u - 19) ≤ -6 + 2u
Distribute left side:
→ -5u + 95 ≤ -6 + 2u
Add 5u to both sides:
→ 95 ≤ -6 + 7u
Add 6:
→ 101 ≤ 7u
Divide by 7:
→ u ≥ 101/7 ≈ 14.428...
✔ Final Answer for #10:
u ≥ 101/7
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11. 4h - 6 ≥ 140
Add 6:
→ 4h ≥ 146
Divide by 4:
→ h ≥ 146/4 = 73/2 = 36.5
✔ Final Answer for #11:
h ≥ 36.5
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12. 144 ≤ 12a - 4
Add 4:
→ 148 ≤ 12a
Divide by 12:
→ a ≥ 148/12 = 37/3 ≈ 12.333...
✔ Final Answer for #12:
a ≥ 37/3
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13. 22 ≤ -7(b - 19)
Divide both sides by -7 → flip inequality!
→ 22 / (-7) ≥ b - 19
→ -22/7 ≥ b - 19
Add 19:
→ -22/7 + 133/7 ≥ b
→ 111/7 ≥ b
→ b ≤ 111/7 ≈ 15.857...
✔ Final Answer for #13:
b ≤ 111/7
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14. 6u + 6 < u + 29
Subtract u:
→ 5u + 6 < 29
Subtract 6:
→ 5u < 23
Divide by 5:
→ u < 23/5 = 4.6
✔ Final Answer for #14:
u < 4.6
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Final Answer:
1. n ≥ -29/6
2. y ≥ -1/6
3. m ≥ 20
4. y ≤ 42
5. r ≥ 1.5
6. g < -8/3
7. m > 1/4
8. m > 14
9. w ≤ 4
10. u ≥ 101/7
11. h ≥ 36.5
12. a ≥ 37/3
13. b ≤ 111/7
14. u < 4.6
Parent Tip: Review the logic above to help your child master the concept of solving inequalities worksheet pdf.
