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.\\(5x + 2 < -3\\) or \\(2x - 6 geq -12\\)\\(\\)\\(\\)what is the correct graph of \\(5x + 2 < -3\\)?\\(\\)\\(\\)a. \\(\\)\\(\\)b. \\(\\)\\(\\)c. \\(\\)\\(\\)d. \\(\\)\\(\\)what is the correct graph of \\(2x - 6 geq -12\\)?\\(\\)\\(\\)a. \\(\\)\\(\\)b. \\(\\)\\(\\)c. \\(\\)\\(\\)d. \\(\\)

Explanation:

Step1: Solve $5x+2 < -3$

Subtract 2 from both sides:
$5x < -3 - 2$
$5x < -5$
Divide by 5:
$x < -1$
This corresponds to a number line with an open arrow pointing left from -1, which is option D.

Step2: Solve $2x-6 \geq -12$

Add 6 to both sides:
$2x \geq -12 + 6$
$2x \geq -6$
Divide by 2:
$x \geq -3$
This corresponds to a number line with a closed bracket pointing right from -3, which is option D.

Step3: Find compound solution

The compound inequality is "or", so combine $x < -1$ and $x \geq -3$. The combined solution set is all real numbers, since every real number satisfies one of the two inequalities. In interval notation: $(-\infty, \infty)$

Answer:

1. Correct graph for $5x+2 < -3$: D. <graph with open arrow left from -1>
2. Correct graph for $2x-6 \geq -12$: D. <graph with closed bracket right from -3>
3. Solution set of compound inequality: $(-\infty, \infty)$