Sure, let's break down the compound inequality [tex]\(-22 > -5x - 7 \geq -3\)[/tex] to find its equivalent form.
Step 1: Split the compound inequality into two separate inequalities.
1. [tex]\(-22 > -5x - 7\)[/tex]
2. [tex]\(-5x - 7 \geq -3\)[/tex]
Step 2: Solve each inequality separately.
### Inequality 1: [tex]\(-22 > -5x - 7\)[/tex]
1. Add 7 to both sides:
[tex]\[
-22 + 7 > -5x
\][/tex]
[tex]\[
-15 > -5x
\][/tex]
2. Divide both sides by [tex]\(-5\)[/tex]. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[
\frac{-15}{-5} < x
\][/tex]
[tex]\[
3 < x
\][/tex]
This can be written as:
[tex]\[
x > 3
\][/tex]
### Inequality 2: [tex]\(-5x - 7 \geq -3\)[/tex]
1. Add 7 to both sides:
[tex]\[
-5x - 7 + 7 \geq -3 + 7
\][/tex]
[tex]\[
-5x \geq 4
\][/tex]
2. Divide both sides by [tex]\(-5\)[/tex]. Again, remember to reverse the inequality sign:
[tex]\[
\frac{-5x}{-5} \leq \frac{4}{-5}
\][/tex]
[tex]\[
x \leq -\frac{4}{5}
\][/tex]
Step 3: Combine the simplified inequalities to represent the solution.
- The solution to the first inequality is [tex]\(x > 3\)[/tex].
- The solution to the second inequality is [tex]\(x \leq -\frac{4}{5}\)[/tex].
So, the combined form, using the correct inequality symbols and line thresholds is:
[tex]\[
-5x - 7 < -22 \quad \text{and} \quad -5x - 7 \geq -3
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-5x-7 < -22 \text{ and } -5x-7 \geq -3}
\][/tex]
