To solve the compound inequality [tex]\(-33 > -3x - 6 \geq -36\)[/tex], it is convenient to first break it down into two separate inequalities and then solve each one.
The compound inequality can be separated as:
1. [tex]\(-33 > -3x - 6\)[/tex]
2. [tex]\(-3x - 6 \geq -36\)[/tex]
#### Step-by-Step Solution for Each Inequality:
1. Solving [tex]\(-33 > -3x - 6\)[/tex]:
[tex]\[
-33 > -3x - 6
\][/tex]
Add [tex]\(6\)[/tex] to both sides:
[tex]\[
-33 + 6 > -3x
\][/tex]
[tex]\[
-27 > -3x
\][/tex]
Divide both sides by [tex]\(-3\)[/tex], remembering to reverse the inequality sign:
[tex]\[
\frac{-27}{-3} < x
\][/tex]
[tex]\[
9 < x \quad \text{or} \quad x > 9
\][/tex]
2. Solving [tex]\(-3x - 6 \geq -36\)[/tex]:
[tex]\[
-3x - 6 \geq -36
\][/tex]
Add [tex]\(6\)[/tex] to both sides:
[tex]\[
-3x - 6 + 6 \geq -36 + 6
\][/tex]
[tex]\[
-3x \geq -30
\][/tex]
Divide both sides by [tex]\(-3\)[/tex], remembering to reverse the inequality sign:
[tex]\[
\frac{-30}{-3} \leq x
\][/tex]
[tex]\[
10 \leq x \quad \text{or} \quad x \geq 10
\][/tex]
#### Combining the Two Results:
From solving both inequalities, we have:
1. [tex]\(x > 9\)[/tex]
2. [tex]\(x \geq 10\)[/tex]
The solution we get must satisfy both of these conditions simultaneously. The overlap of these two intervals is [tex]\(x \geq 10\)[/tex].
However, the question was to find an equivalent form of the compound inequality. The equivalent compound inequalities, before solving further, are:
1. [tex]\(-3x - 6 > -33\)[/tex]
2. [tex]\(-3x - 6 \geq -36\)[/tex]
Thus, among the given choices:
- [tex]\(-3x - 6 > -33\)[/tex] and [tex]\(-3x - 6 \geq -36\)[/tex]
- [tex]\(-3x - 6 < -33\)[/tex] and [tex]\(-3x - 6 \geq -36\)[/tex]
- [tex]\(-3x > -33\)[/tex] and [tex]\(-6 \geq -36\)[/tex]
- [tex]\(-3x - 6 < -33\)[/tex] and [tex]\(-3x - 6 \leq -36\)[/tex]
The correct equivalent form of the compound inequality [tex]\(-33 > -3x - 6 \geq -36\)[/tex] is:
[tex]\[
-3x - 6 > -33 \quad \text{and} \quad -3x - 6 \geq -36
\][/tex]
Thus, the answer is:
[tex]\[
-3x - 6 > -33 \quad \text{and} \quad -3x - 6 \geq -36
\][/tex]
