To determine which compound inequality the number line represents, we need to analyze each of the provided inequalities step by step.
1. Inequality: [tex]\(5x \geq -15\)[/tex] or [tex]\(5x < 10\)[/tex]
- To solve [tex]\(5x \geq -15\)[/tex]:
[tex]\[
x \geq \frac{-15}{5} \quad \Rightarrow \quad x \geq -3
\][/tex]
- To solve [tex]\(5x < 10\)[/tex]:
[tex]\[
x < \frac{10}{5} \quad \Rightarrow \quad x < 2
\][/tex]
So, the combined inequality is:
[tex]\[
x \geq -3 \quad \text{or} \quad x < 2
\][/tex]
2. Inequality: [tex]\(2x \leq -6\)[/tex] or [tex]\(2x \geq 4\)[/tex]
- To solve [tex]\(2x \leq -6\)[/tex]:
[tex]\[
x \leq \frac{-6}{2} \quad \Rightarrow \quad x \leq -3
\][/tex]
- To solve [tex]\(2x \geq 4\)[/tex]:
[tex]\[
x \geq \frac{4}{2} \quad \Rightarrow \quad x \geq 2
\][/tex]
So, the combined inequality is:
[tex]\[
x \leq -3 \quad \text{or} \quad x \geq 2
\][/tex]
3. Inequality: [tex]\(-13 \leq 4x - 1 < 7\)[/tex]
- To isolate [tex]\(x\)[/tex], first add 1 to all parts:
[tex]\[
-13 + 1 \leq 4x - 1 + 1 < 7 + 1 \quad \Rightarrow \quad -12 \leq 4x < 8
\][/tex]
- Next, divide all parts by 4:
[tex]\[
\frac{-12}{4} \leq \frac{4x}{4} < \frac{8}{4} \quad \Rightarrow \quad -3 \leq x < 2
\][/tex]
So, the inequality simplifies to:
[tex]\[
-3 \leq x < 2
\][/tex]
4. Inequality: [tex]\(-4x \geq 12\)[/tex] and [tex]\(-4x < -8\)[/tex]
- To solve [tex]\(-4x \geq 12\)[/tex]:
[tex]\[
x \leq \frac{12}{-4} \quad \Rightarrow \quad x \leq -3 \quad \text{(flip the inequality sign)}
\][/tex]
- To solve [tex]\(-4x < -8\)[/tex]:
[tex]\[
x > \frac{-8}{-4} \quad \Rightarrow \quad x > 2 \quad \text{(flip the inequality sign)}
\][/tex]
So, the combined inequality is:
[tex]\[
x \leq -3 \quad \text{and} \quad x > 2
\][/tex]
After evaluating each of these inequalities, we see that the inequality that matches a proper range on the number line is:
[tex]\[
-3 \leq x < 2
\][/tex]
Therefore, the combined inequality that the number line represents is:
[tex]\[
-13 \leq 4x - 1 < 7
\][/tex]
Thus, the answer is [tex]\(3\)[/tex].
