1. Solve $2(x - 3) < 4$: Expand: $2x - 6 < 4$ Add 6: $2x < 10$ Divide by 2: $x < 5$ Answer: (c) $x < 5$ 2. Solve $3 - 2x \leq 6$: Subtract 3: $-2x \leq 3$ Divide by -2 and reverse inequality: $x \geq -\frac{3}{2}$ Answer: (a) $x \geq -\frac{3}{2}$ 3. Solve $2(x - 4) - 3 > 2x - 1$: Expand left: $2x - 8 -3 > 2x - 1$ Simplify: $2x - 11 > 2x - 1$ Subtract $2x$: $-11 > -1$ which is false Answer: (a) $\emptyset$ (no solution) 4. Solve $2(x - 4) + 11 > 2x - 1$: Expand: $2x - 8 + 11 > 2x - 1$ Simplify: $2x + 3 > 2x -1$ Subtract $2x$: $3 > -1$ true for all $x$ Answer: (d) $(-\infty , \infty)$ 5. The interval $[2,5]$ means $x$ includes 2 and 5: Answer: (b) $2 \leq x \leq 5$ 6. The solution $[-6, \infty)$ means $x$ starts at -6 including it and extends to the right: Answer: (d) a filled circle at -6 with a right arrow 7. Solve compound $-x + 3 \leq 2x + 3 \leq 9$: Split: $-x + 3 \leq 2x + 3$ and $2x + 3 \leq 9$ First inequality: $-x + 3 \leq 2x +3$ Subtract 3: $-x \leq 2x$ Add $x$: $0 \leq 3x$ Divide by 3: $0 \leq x$ Second inequality: $2x + 3 \leq 9$ Subtract 3: $2x \leq 6$ Divide by 2: $x \leq 3$ Combine: $0 \leq x \leq 3$ Answer: (b) $0 \leq x \leq 3$ 8. Solve $-4 \leq 2x + 2 \leq 10$: Split inequalities: $-4 \leq 2x + 2$ $2x + 2 \leq 10$ First: Subtract 2: $-6 \leq 2x$ Divide by 2: $-3 \leq x$ Second: Subtract 2: $2x \leq 8$ Divide by 2: $x \leq 4$ Combine: $-3 \leq x \leq 4$ Answer: (c) $-3 \leq x \leq 4$ 9. Solve $4 \leq -x + 3 \leq 12$: Split: $4 \leq -x + 3$ $-x + 3 \leq 12$ First: Subtract 3: $1 \leq -x$ Multiply by -1 and reverse inequality: $-1 \geq x$ or $x \leq -1$ Second: Subtract 3: $-x \leq 9$ Multiply by -1 and reverse inequality: $x \geq -9$ Combine: $-9 \leq x \leq -1$ Answer: (d) $-9 \leq x \leq -1$ 26. Solve $7 - \frac{2}{3}x < x - 8$: Bring terms: $7 + 8 < x + \frac{2}{3} x$ $15 < \frac{5}{3} x$ Multiply by $\frac{3}{5}$: $x > 9$ Answer: (a) $x > 9$