Question

solve each compound inequality and graph its solution.

1. $m - 2 < -8$ or $dfrac{m}{8} > 1$
2. $-1 < 9 + n < 17$
3. $2x < 10$ or $dfrac{x}{2} geq 3$
4. $x + 8 geq 9$ and $dfrac{x}{7} leq 1$
5. $-3 leq dfrac{p}{2} < 0$
6. $r + 5 geq 12$ or $dfrac{r}{9} < 0$
7. $7v - 5 geq 65$ or $-3v - 2 geq -2$
8. $-10b + 3 leq -37$ or $3b - 10 leq -25$

Explanation:

Problem 1: \( m - 2 < -8 \) or \( \frac{m}{8} > 1 \)

Step 1: Solve \( m - 2 < -8 \)

Add 2 to both sides: \( m - 2 + 2 < -8 + 2 \)
\( m < -6 \)

Step 2: Solve \( \frac{m}{8} > 1 \)

Multiply both sides by 8: \( \frac{m}{8} \times 8 > 1 \times 8 \)
\( m > 8 \)

Step 1: Subtract 9 from all parts

\( -1 - 9 < 9 + n - 9 < 17 - 9 \)
\( -10 < n < 8 \)

Step 1: Solve \( 2x < 10 \)

Divide both sides by 2: \( \frac{2x}{2} < \frac{10}{2} \)
\( x < 5 \)

Step 2: Solve \( \frac{x}{2} \geq 3 \)

Multiply both sides by 2: \( \frac{x}{2} \times 2 \geq 3 \times 2 \)
\( x \geq 6 \)

Answer:

\( m < -6 \) or \( m > 8 \)

Problem 2: \( -1 < 9 + n < 17 \)