Let’s solve each inequality one by one. We’ll isolate the variable (like x or a) on one side, just like solving equations — but remember: if you multiply or divide by a negative number, flip the inequality sign! (None of these need that though.)
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1. 3x - 8 ≤ 22
Add 8 to both sides:
3x ≤ 30
Divide by 3:
x ≤ 10
✔ Final answer for #1:
x ≤ 10
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2. 5r + 10 ≥ 19
Subtract 10 from both sides:
5r ≥ 9
Divide by 5:
r ≥ 9/5 → r ≥ 1.8
✔ Final answer for #2:
r ≥ 1.8
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3. 6x - 7 ≤ 5
Add 7 to both sides:
6x ≤ 12
Divide by 6:
x ≤ 2
✔ Final answer for #3:
x ≤ 2
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4. -t + 1 ≥ 6
Subtract 1 from both sides:
-t ≥ 5
Multiply both sides by -1 → FLIP the sign!
t ≤ -5
✔ Final answer for #4:
t ≤ -5
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5. 5a - 4 > 2a + 8
Subtract 2a from both sides:
3a - 4 > 8
Add 4 to both sides:
3a > 12
Divide by 3:
a > 4
✔ Final answer for #5:
a > 4
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6. x + 1 ≥ 3x - 18
Subtract x from both sides:
1 ≥ 2x - 18
Add 18 to both sides:
19 ≥ 2x
Divide by 2:
9.5 ≥ x → or x ≤ 9.5
✔ Final answer for #6:
x ≤ 9.5
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7. 5x - 2 ≥ 68
Add 2 to both sides:
5x ≥ 70
Divide by 5:
x ≥ 14
✔ Final answer for #7:
x ≥ 14
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8. 5(x - 3) ≥ 40
First, divide both sides by 5:
x - 3 ≥ 8
Add 3 to both sides:
x ≥ 11
✔ Final answer for #8:
x ≥ 11
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9. 4x + 7 < 11
Subtract 7 from both sides:
4x < 4
Divide by 4:
x < 1
✔ Final answer for #9:
x < 1
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10. 4(x - 2) > -20
Divide both sides by 4:
x - 2 > -5
Add 2 to both sides:
x > -3
✔ Final answer for #10:
x > -3
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11. 4 - 2x ≥ -8
Subtract 4 from both sides:
-2x ≥ -12
Divide by -2 → FLIP the sign!
x ≤ 6
✔ Final answer for #11:
x ≤ 6
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12. 7x - 5 ≥ 3x + 11
Subtract 3x from both sides:
4x - 5 ≥ 11
Add 5 to both sides:
4x ≥ 16
Divide by 4:
x ≥ 4
✔ Final answer for #12:
x ≥ 4
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Final Answer:
1. x ≤ 10
2. r ≥ 1.8
3. x ≤ 2
4. t ≤ -5
5. a > 4
6. x ≤ 9.5
7. x ≥ 14
8. x ≥ 11
9. x < 1
10. x > -3
11. x ≤ 6
12. x ≥ 4
Parent Tip: Review the logic above to help your child master the concept of inequality worksheet 6th grade.
