Solving Inequalities Worksheet - Teach-nology - Free Printable

Let's solve each inequality step by step and provide the solution for every problem on the "Solving Inequalities Worksheet".

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1. $ 6(n + 10) \geq 31 $

Step 1: Distribute
$ 6n + 60 \geq 31 $

Step 2: Subtract 60 from both sides
$ 6n \geq 31 - 60 $
$ 6n \geq -29 $

Step 3: Divide by 6
$ n \geq -\frac{29}{6} $
$ n \geq -4\frac{5}{6} $

✔ Answer: $ n \geq -\frac{29}{6} $

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2. $ 6(y + 6) \geq 35 $

Step 1: Distribute
$ 6y + 36 \geq 35 $

Step 2: Subtract 36
$ 6y \geq -1 $

Step 3: Divide by 6
$ y \geq -\frac{1}{6} $

✔ Answer: $ y \geq -\frac{1}{6} $

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3. $ -8 \leq \frac{m}{10} - 10 $

Step 1: Add 10 to both sides
$ -8 + 10 \leq \frac{m}{10} $
$ 2 \leq \frac{m}{10} $

Step 2: Multiply both sides by 10
$ 20 \leq m $
or
$ m \geq 20 $

✔ Answer: $ m \geq 20 $

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4. $ 5 \leq \frac{v}{-3} + 19 $

Step 1: Subtract 19 from both sides
$ 5 - 19 \leq \frac{v}{-3} $
$ -14 \leq \frac{v}{-3} $

Step 2: Multiply both sides by $-3$. Remember: When multiplying or dividing by a negative number, reverse the inequality sign.
$ (-14)(-3) \geq v $
$ 42 \geq v $
or
$ v \leq 42 $

✔ Answer: $ v \leq 42 $

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5. $ 6r + 10 \geq 19 $

Step 1: Subtract 10
$ 6r \geq 9 $

Step 2: Divide by 6
$ r \geq \frac{9}{6} = \frac{3}{2} $

✔ Answer: $ r \geq \frac{3}{2} $

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6. $ 7g + 18 < 4g + 10 $

Step 1: Subtract $4g$ from both sides
$ 3g + 18 < 10 $

Step 2: Subtract 18
$ 3g < -8 $

Step 3: Divide by 3
$ g < -\frac{8}{3} $

✔ Answer: $ g < -\frac{8}{3} $

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7. $ 5m + 2 < 9m + 1 $

Step 1: Subtract $5m$ from both sides
$ 2 < 4m + 1 $

Step 2: Subtract 1
$ 1 < 4m $

Step 3: Divide by 4
$ \frac{1}{4} < m $
or
$ m > \frac{1}{4} $

✔ Answer: $ m > \frac{1}{4} $

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8. $ -21 > -2m + 7 $

Step 1: Subtract 7
$ -21 - 7 > -2m $
$ -28 > -2m $

Step 2: Divide by $-2$, reverse inequality
$ \frac{-28}{-2} < m $
$ 14 < m $
or
$ m > 14 $

✔ Answer: $ m > 14 $

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9. $ 58 \geq 14 + 11w $

Step 1: Subtract 14
$ 44 \geq 11w $

Step 2: Divide by 11
$ 4 \geq w $
or
$ w \leq 4 $

✔ Answer: $ w \leq 4 $

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10. $ -5(u - 19) \leq -6 + 2u $

Step 1: Distribute
$ -5u + 95 \leq -6 + 2u $

Step 2: Move all terms with $u$ to one side: subtract $2u$
$ -7u + 95 \leq -6 $

Step 3: Subtract 95
$ -7u \leq -101 $

Step 4: Divide by $-7$, reverse inequality
$ u \geq \frac{101}{7} $
$ u \geq 14\frac{3}{7} $

✔ Answer: $ u \geq \frac{101}{7} $

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11. $ 4h - 6 \geq 140 $

Step 1: Add 6
$ 4h \geq 146 $

Step 2: Divide by 4
$ h \geq \frac{146}{4} = \frac{73}{2} = 36.5 $

✔ Answer: $ h \geq \frac{73}{2} $

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12. $ 144 \leq 12a - 4 $

Step 1: Add 4
$ 148 \leq 12a $

Step 2: Divide by 12
$ \frac{148}{12} \leq a $
$ \frac{37}{3} \leq a $
$ a \geq 12\frac{1}{3} $

✔ Answer: $ a \geq \frac{37}{3} $

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13. $ 22 \leq -7(b - 19) $

Step 1: Distribute
$ 22 \leq -7b + 133 $

Step 2: Subtract 133
$ 22 - 133 \leq -7b $
$ -111 \leq -7b $

Step 3: Divide by $-7$, reverse inequality
$ \frac{-111}{-7} \geq b $
$ \frac{111}{7} \geq b $
$ b \leq 15\frac{6}{7} $

✔ Answer: $ b \leq \frac{111}{7} $

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14. $ 6u + 6 < u + 29 $

Step 1: Subtract $u$
$ 5u + 6 < 29 $

Step 2: Subtract 6
$ 5u < 23 $

Step 3: Divide by 5
$ u < \frac{23}{5} = 4.6 $

✔ Answer: $ u < \frac{23}{5} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ n \geq -\frac{29}{6} $ |
| 2 | $ y \geq -\frac{1}{6} $ |
| 3 | $ m \geq 20 $ |
| 4 | $ v \leq 42 $ |
| 5 | $ r \geq \frac{3}{2} $ |
| 6 | $ g < -\frac{8}{3} $ |
| 7 | $ m > \frac{1}{4} $ |
| 8 | $ m > 14 $ |
| 9 | $ w \leq 4 $ |
| 10 | $ u \geq \frac{101}{7} $ |
| 11 | $ h \geq \frac{73}{2} $ |
| 12 | $ a \geq \frac{37}{3} $ |
| 13 | $ b \leq \frac{111}{7} $ |
| 14 | $ u < \frac{23}{5} $ |

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🔍 Key Tips:

- Always isolate the variable.
- Remember: When multiplying or dividing by a negative number, flip the inequality sign.
- Simplify fractions when possible.

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