1. **Problem 7: Solve $x + 7 > 18$** We want to find all $x$ such that $x + 7 > 18$. 2. **Formula and rules:** To isolate $x$, subtract 7 from both sides: $$x + 7 > 18$$ $$x + \cancel{7} - \cancel{7} > 18 - 7$$ 3. **Simplify:** $$x > 11$$ 4. **Answer:** The solution set is all $x$ such that $x > 11$. --- 1. **Problem 8: Solve $-2x \leq 16$** 2. **Formula and rules:** Divide both sides by $-2$. Remember, dividing by a negative number reverses the inequality sign: $$-2x \leq 16$$ $$\frac{-2x}{-2} \geq \frac{16}{-2}$$ 3. **Simplify:** $$x \geq -8$$ 4. **Answer:** The solution set is all $x$ such that $x \geq -8$. --- 1. **Problem 9: Solve $\frac{1}{3}x \geq 4$** 2. **Formula and rules:** Multiply both sides by 3 to clear the fraction: $$\frac{1}{3}x \geq 4$$ $$3 \times \frac{1}{3}x \geq 3 \times 4$$ 3. **Simplify:** $$x \geq 12$$ 4. **Answer:** The solution set is all $x$ such that $x \geq 12$. --- 1. **Problem 10: Solve $20 < 3x + 5$** 2. **Formula and rules:** Subtract 5 from both sides: $$20 < 3x + 5$$ $$20 - 5 < 3x + 5 - 5$$ 3. **Simplify:** $$15 < 3x$$ Divide both sides by 3: $$\frac{15}{3} < \frac{3x}{3}$$ 4. **Simplify:** $$5 < x$$ 5. **Answer:** The solution set is all $x$ such that $x > 5$. --- 1. **Problem 11: Solve compound inequality $4 \leq \frac{1}{2}x < 5$** 2. **Formula and rules:** Multiply all parts by 2 to clear the fraction: $$4 \leq \frac{1}{2}x < 5$$ $$2 \times 4 \leq 2 \times \frac{1}{2}x < 2 \times 5$$ 3. **Simplify:** $$8 \leq x < 10$$ 4. **Answer:** The solution set is all $x$ such that $8 \leq x < 10$. --- 1. **Problem 12: Solve compound inequality $3 < 4 - 6x < 14$** 2. **Formula and rules:** Subtract 4 from all parts: $$3 - 4 < 4 - 6x - 4 < 14 - 4$$ 3. **Simplify:** $$-1 < -6x < 10$$ Divide all parts by $-6$ and reverse inequalities: $$\frac{-1}{-6} > x > \frac{10}{-6}$$ 4. **Simplify fractions:** $$\frac{1}{6} > x > -\frac{5}{3}$$ Rewrite in standard form: $$-\frac{5}{3} < x < \frac{1}{6}$$ 5. **Answer:** The solution set is all $x$ such that $-\frac{5}{3} < x < \frac{1}{6}$. --- 1. **Problem 13: Check if $x=5$ satisfies $-2x + 5 \geq 7$** 2. **Substitute $x=5$:** $$-2(5) + 5 \geq 7$$ $$-10 + 5 \geq 7$$ $$-5 \geq 7$$ 3. **Evaluate:** $-5$ is not greater than or equal to $7$, so the inequality is false. 4. **Answer:** $x=5$ does not satisfy the inequality. --- 1. **Problem 14: Check if $x=6$ satisfies $23 \leq 6x + 8 < 44$** 2. **Substitute $x=6$:** $$23 \leq 6(6) + 8 < 44$$ $$23 \leq 36 + 8 < 44$$ $$23 \leq 44 < 44$$ 3. **Evaluate:** $44 < 44$ is false (44 is not less than 44), so the compound inequality is false. 4. **Answer:** $x=6$ does not satisfy the inequality. --- 1. **Problem 15: Check if $x=\frac{7}{4}$ satisfies $-\frac{1}{4} < \frac{1}{8}x - 2 \leq \frac{5}{4}$** 2. **Substitute $x=\frac{7}{4}$:** $$-\frac{1}{4} < \frac{1}{8} \times \frac{7}{4} - 2 \leq \frac{5}{4}$$ 3. **Calculate:** $$\frac{1}{8} \times \frac{7}{4} = \frac{7}{32}$$ So: $$-\frac{1}{4} < \frac{7}{32} - 2 \leq \frac{5}{4}$$ Convert $-2$ to $\frac{-64}{32}$: $$-\frac{1}{4} < \frac{7}{32} - \frac{64}{32} \leq \frac{5}{4}$$ Simplify: $$-\frac{1}{4} < -\frac{57}{32} \leq \frac{5}{4}$$ 4. **Evaluate:** $-\frac{1}{4} = -0.25$, $-\frac{57}{32} \approx -1.78125$, $\frac{5}{4} = 1.25$ Check inequalities: $$-0.25 < -1.78125$$ is false. 5. **Answer:** $x=\frac{7}{4}$ does not satisfy the inequality. --- **Summary of solutions:** - 7) $x > 11$ - 8) $x \geq -8$ - 9) $x \geq 12$ - 10) $x > 5$ - 11) $8 \leq x < 10$ - 12) $-\frac{5}{3} < x < \frac{1}{6}$ - 13) $x=5$ does not satisfy - 14) $x=6$ does not satisfy - 15) $x=\frac{7}{4}$ does not satisfy