To determine which compound inequality is represented by the number line, let's analyze each of the provided inequalities and the intervals they imply.
1. [tex]\(2x \leq -6\)[/tex] or [tex]\(2x \geq 4\)[/tex]
- Solve [tex]\(2x \leq -6\)[/tex]:
[tex]\[
x \leq -3
\][/tex]
- Solve [tex]\(2x \geq 4\)[/tex]:
[tex]\[
x \geq 2
\][/tex]
The number line representation for this inequality will have two distinct intervals:
[tex]\[
x \leq -3 \quad \text{or} \quad x \geq 2.
\][/tex]
2. [tex]\(-13 \leq 4x - 1 < 7\)[/tex]
- Solve [tex]\(-13 \leq 4x - 1\)[/tex]:
[tex]\[
-12 \leq 4x \quad \Rightarrow \quad -3 \leq x
\][/tex]
- Solve [tex]\(4x - 1 < 7\)[/tex]:
[tex]\[
4x < 8 \quad \Rightarrow \quad x < 2
\][/tex]
The number line representation for this inequality will be a single interval:
[tex]\[
-3 \leq x < 2.
\][/tex]
3. [tex]\(-4x \geq 12\)[/tex] and [tex]\(-4x < -8\)[/tex]
- Solve [tex]\(-4x \geq 12\)[/tex]:
[tex]\[
x \leq -3
\][/tex]
- Solve [tex]\(-4x < -8\)[/tex]:
[tex]\[
x > 2
\][/tex]
The number line representation for this inequality will be:
[tex]\[
x \leq -3 \quad \text{and} \quad x > 2.
\][/tex]
This inequality is actually impossible to satisfy simultaneously as there are no values for [tex]\(x\)[/tex] that can be both less than [tex]\(-3\)[/tex] and greater than [tex]\(2\)[/tex].
4. [tex]\(5x \geq -15\)[/tex] or [tex]\(5x < 10\)[/tex]
- Solve [tex]\(5x \geq -15\)[/tex]:
[tex]\[
x \geq -3
\][/tex]
- Solve [tex]\(5x < 10\)[/tex]:
[tex]\[
x < 2
\][/tex]
The number line representation for this inequality will cover almost the entire range:
[tex]\[
x \geq -3 \quad \text{or} \quad x < 2.
\][/tex]
To summarize:
- The intervals for the inequalities are:
- [tex]\(2x \leq -6\)[/tex] or [tex]\(2x \geq 4\)[/tex]: [tex]\(x \leq -3\)[/tex] or [tex]\(x \geq 2\)[/tex].
- [tex]\(-13 \leq 4x - 1 < 7\)[/tex]: [tex]\(-3 \leq x < 2\)[/tex].
- [tex]\(-4x \geq 12\)[/tex] and [tex]\(-4x < -8\)[/tex]: [tex]\(x \leq -3\)[/tex] and [tex]\(x > 2\)[/tex] (which cannot happen).
- [tex]\(5x \geq -15\)[/tex] or [tex]\(5x < 10\)[/tex]: [tex]\(x \geq -3\)[/tex] or [tex]\(x < 2\)[/tex] (which means it covers virtually all [tex]\(x\)[/tex]).
From the analysis, it seems that [tex]\(-3 \leq x < 2\)[/tex] is correctly represented by the number line spanning from [tex]\(-3\)[/tex] to [tex]\(2\)[/tex] without including [tex]\(2\)[/tex].
Therefore, the compound inequality that matches our number line representation is:
[tex]\[
-13 \leq 4x - 1 < 7.
\][/tex]
