To solve the given compound inequality [tex]\(-22 > -5x - 7 \geq -3\)[/tex], we need to break it down into two separate inequalities and analyze each part:
1. [tex]\(-22 > -5x - 7\)[/tex]
2. [tex]\(-5x - 7 \geq -3\)[/tex]
Step 1: Solve the inequality [tex]\(-22 > -5x - 7\)[/tex]
- Start by isolating the term with [tex]\(x\)[/tex]. To do this, add 7 to both sides of the inequality:
[tex]\[
-22 + 7 > -5x - 7 + 7
\][/tex]
[tex]\[
-15 > -5x
\][/tex]
- Next, divide both sides by -5. Remember, dividing by a negative number reverses the inequality:
[tex]\[
\frac{-15}{-5} < x
\][/tex]
[tex]\[
3 < x \quad \text{or} \quad x > 3
\][/tex]
Step 2: Solve the inequality [tex]\(-5x - 7 \geq -3\)[/tex]
- Again, isolate the term with [tex]\(x\)[/tex] by adding 7 to both sides of the inequality:
[tex]\[
-5x - 7 + 7 \geq -3 + 7
\][/tex]
[tex]\[
-5x \geq 4
\][/tex]
- Next, divide both sides by -5. Remember, dividing by a negative number reverses the inequality:
[tex]\[
\frac{-5x}{-5} \leq \frac{4}{-5}
\][/tex]
[tex]\[
x \leq -0.8
\][/tex]
Combining the results:
The compound inequality [tex]\(-22 > -5x - 7 \geq -3\)[/tex] is equivalent to:
[tex]\[
x > 3 \quad \text{and} \quad x \leq -0.8
\][/tex]
These results are contradictory for real numbers because a number cannot be both greater than 3 and less than or equal to -0.8 at the same time.
Now, examine the provided choices to determine which one expresses the same operations we performed:
1. [tex]\(-5x - 7 < -22\)[/tex] and [tex]\(-5x - 7 \geq -3\)[/tex]
2. [tex]\(-5x - 7 > -22\)[/tex] and [tex]\(-5x - 7 \geq -3\)[/tex]
3. [tex]\(-5x > -22\)[/tex] and [tex]\(-7 \geq -3\)[/tex]
4. [tex]\(-5x - 7 < -22\)[/tex] and [tex]\(-5x - 7 \leq -3\)[/tex]
Analysis of choices:
- From our work:
- The first part [tex]\(-22 > -5x - 7\)[/tex] was rewritten equivalently as [tex]\(-5x - 7 < -22\)[/tex] (same as choice 1 and 4).
- The second part [tex]\(-5x - 7 \geq -3\)[/tex] is already in one of the choices (choice 1 and 2).
We see the inequality [tex]\(-5x - 7 < -22\)[/tex] and [tex]\(-5x - 7 \geq -3\)[/tex] matches our steps exactly.
Thus, the equivalent form from the provided choices is:
[tex]\[
-5x - 7 < -22 \quad \text{and} \quad -5x - 7 \geq -3
\][/tex]
Therefore, the correct answer is:
Option 1: [tex]\(-5x - 7 < -22 \text{ and } -5x - 7 \geq -3\)[/tex].
