Question

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

a) $2x < 6$

b) $-3x - 6 \geq 2(x - 3)$

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

d) $-8 \leq x - 2 < 1$

Explanation:

Part (a)

Step1: Divide both sides by 2

To solve \(2x < 6\), we divide both sides of the inequality by 2 (since 2 is positive, the inequality sign remains the same).
\(\frac{2x}{2} < \frac{6}{2}\)

Step2: Simplify

Simplifying both sides gives us \(x < 3\).

Step1: Expand the right side

First, expand \(2(x - 3)\) using the distributive property: \(2(x - 3)=2x - 6\). So the inequality becomes \(-3x - 6 \geq 2x - 6\).

Step2: Add \(3x\) to both sides

Adding \(3x\) to both sides to get all \(x\) terms on one side: \(-3x - 6 + 3x \geq 2x - 6 + 3x\), which simplifies to \(-6 \geq 5x - 6\).

Step3: Add 6 to both sides

Adding 6 to both sides: \(-6 + 6 \geq 5x - 6 + 6\), which simplifies to \(0 \geq 5x\).

Step4: Divide by 5

Divide both sides by 5 (since 5 is positive, the inequality sign remains the same): \(\frac{0}{5} \geq \frac{5x}{5}\), so \(0 \geq x\) or \(x \leq 0\).

Step1: Subtract 12 from both sides

Subtract 12 from both sides: \(4k + 12 - 12 < 4 - 12\), which simplifies to \(4k < -8\).

Step2: Divide by 4

Divide both sides by 4 (positive, so inequality sign remains): \(\frac{4k}{4} < \frac{-8}{4}\), so \(k < -2\).

Solve \(4 - 2k > 4\)

Answer:

\(x < 3\)

Part (b)