Inequalities Worksheets with Answer Key - Free Printable

Here are the step-by-step solutions for each inequality on your worksheet.

1) $6 > -8 - \frac{-7}{7}$
* First, simplify the fraction: $\frac{-7}{7} = -1$.
* The equation becomes: $6 > -8 - (-1)$.
* Subtracting a negative is the same as adding: $6 > -8 + 1$.
* Simplify the right side: $6 > -7$.
* Since $6$ is always greater than $-7$, this statement is always true.
* Answer: All real numbers (or True)

2) $-2x + 10 \leq 100$
* Subtract $10$ from both sides: $-2x \leq 90$.
* Divide by $-2$. Remember: When you divide or multiply by a negative number, flip the inequality sign.
* $x \geq \frac{90}{-2}$
* Answer: $x \geq -45$

3) $5x - 2 \geq 4 + 3x$
* Subtract $3x$ from both sides to get x's on one side: $2x - 2 \geq 4$.
* Add $2$ to both sides: $2x \geq 6$.
* Divide by $2$: $x \geq 3$.
* Answer: $x \geq 3$

4) $5p - 6 \leq 2p - 7$
* Subtract $2p$ from both sides: $3p - 6 \leq -7$.
* Add $6$ to both sides: $3p \leq -1$.
* Divide by $3$: $p \leq -\frac{1}{3}$.
* Answer: $p \leq -\frac{1}{3}$

5) $8q > q + 2$
* Subtract $q$ from both sides: $7q > 2$.
* Divide by $7$: $q > \frac{2}{7}$.
* Answer: $q > \frac{2}{7}$

6) $2(s + 1) - 3(s - 2) < s + 6$
* Distribute the numbers into the parentheses: $2s + 2 - 3s + 6 < s + 6$.
* *Note: $-3 \times -2 = +6$*
* Combine like terms on the left side ($2s - 3s = -s$ and $2 + 6 = 8$): $-s + 8 < s + 6$.
* Add $s$ to both sides: $8 < 2s + 6$.
* Subtract $6$ from both sides: $2 < 2s$.
* Divide by $2$: $1 < s$ (which is the same as $s > 1$).
* Answer: $s > 1$

7) $p - 2 \geq -5$
* Add $2$ to both sides: $p \geq -5 + 2$.
* Answer: $p \geq -3$

8) $5l + 3 \leq 2l + 2$
* Subtract $2l$ from both sides: $3l + 3 \leq 2$.
* Subtract $3$ from both sides: $3l \leq -1$.
* Divide by $3$: $l \leq -\frac{1}{3}$.
* Answer: $l \leq -\frac{1}{3}$

9) $3(t + \frac{2}{3}) > 5(2t + 5)$
* Distribute the $3$ on the left: $3t + 2 > 5(2t + 5)$.
* *Note: $3 \times \frac{2}{3} = 2$*
* Distribute the $5$ on the right: $3t + 2 > 10t + 25$.
* Subtract $3t$ from both sides: $2 > 7t + 25$.
* Subtract $25$ from both sides: $-23 > 7t$.
* Divide by $7$: $-\frac{23}{7} > t$ (which is the same as $t < -\frac{23}{7}$).
* Answer: $t < -\frac{23}{7}$ (or $t < -3 \frac{2}{7}$)

10) $-22 \leq -7(v - 19)$
* Distribute the $-7$ on the right side: $-22 \leq -7v + 133$.
* *Note: $-7 \times -19 = +133$*
* Subtract $133$ from both sides: $-155 \leq -7v$.
* Divide by $-7$. Remember: Flip the inequality sign because you are dividing by a negative.
* $\frac{-155}{-7} \geq v$
* $\frac{155}{7} \geq v$ (which is the same as $v \leq \frac{155}{7}$)
* Answer: $v \leq \frac{155}{7}$ (or $v \leq 22 \frac{1}{7}$)

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Final Answer:
1) All real numbers
2) $x \geq -45$
3) $x \geq 3$
4) $p \leq -\frac{1}{3}$
5) $q > \frac{2}{7}$
6) $s > 1$
7) $p \geq -3$
8) $l \leq -\frac{1}{3}$
9) $t < -\frac{23}{7}$
10) $v \leq \frac{155}{7}$