Question

22 (a - d) solve the inequalities.
show all work.
graph on the number line.

a) 2x < 6

b) -3x - 6 ≥ 2(x - 3)

c) 4k + 12 < 4 or 4 - 2k > 4

d) -8 ≤ x - 2 < 1

Explanation:

Part a: Solve $2x < 6$

Step1: Divide both sides by 2

$\frac{2x}{2} < \frac{6}{2}$

Step2: Simplify

$x < 3$

Part b: Solve $-3x - 6 \geq 2(x - 3)$

Step1: Expand right side

$-3x - 6 \geq 2x - 6$

Step2: Add $3x$ to both sides

$-6 \geq 5x - 6$

Step3: Add 6 to both sides

$0 \geq 5x$

Step4: Divide by 5

$x \leq 0$

Part c: Solve $4k + 12 < 4$ or $4 - 2k > 4$

Step1: Solve first inequality

$4k < -8 \implies k < -2$

Step2: Solve second inequality

$-2k > 0 \implies k < 0$

Step3: Combine with "or"

$k < 0$

Part d: Solve $-8 \leq x - 2 < 1$

Step1: Add 2 to all parts

$-8 + 2 \leq x < 1 + 2$

Step2: Simplify

$-6 \leq x < 3$

Note: For graphs:

• a) Open circle at 3, shade left.
• b) Closed circle at 0, shade left.
• c) Open circle at 0, shade left.
• d) Closed circle at -6, open at 3, shade between.

Answer:

a) $x < 3$
b) $x \leq 0$
c) $k < -2$ or $k < 0$ (simplified to $k < 0$)
d) $-6 \leq x < 3$