Question

solve the compound inequality. use graphs to show the solution set to each of the two given inequalities, as well as a third graph that shows the solution set of the compound inequality. express the solution set in interval notation.
3x + 1 < -2 or 2x - 8 ≥ -18

what is the correct graph of 3x + 1 < -2?
a.
<---=====-10 0 10--->

b.
<---(-10 0 10--->

c.
<---=====-10 0 10--->

d.
<---)=====-10 0 10--->

what is the correct graph of 2x - 8 ≥ -18?
a.
<---=====-10 0 10--->

b.
<---=====-10 0 10--->

c.
<---(=====-10 0 10--->

d.
<---)=====-10 0 10--->

Explanation:

Step1: Solve $3x+1 < -2$

Subtract 1 from both sides:
$3x + 1 - 1 < -2 - 1$
$3x < -3$
Divide by 3:
$x < -1$
This corresponds to a number line with an open circle at $-1$ and shading to the left, which is option D.

Step2: Solve $2x-8 \geq -18$

Add 8 to both sides:
$2x - 8 + 8 \geq -18 + 8$
$2x \geq -10$
Divide by 2:
$x \geq -5$
This corresponds to a number line with a closed circle at $-5$ and shading to the right, which is option A.

Step3: Find compound solution

The compound inequality is "or", so combine the solution sets: $x < -1$ or $x \geq -5$, which covers all real numbers.
Interval notation: $(-\infty, \infty)$

Answer:

1. Correct graph for $3x+1 < -2$: D. <graph with open circle at -1, shading left>
2. Correct graph for $2x-8 \geq -18$: A. <graph with closed circle at -5, shading right>
3. Compound solution set in interval notation: $(-\infty, \infty)$