To solve the compound inequality [tex]\(-2x + 9 \leq 13\)[/tex] or [tex]\(-3x - 1 > 8\)[/tex], let's break it down into two separate inequalities and solve each step-by-step.
### Solving the first inequality: [tex]\(-2x + 9 \leq 13\)[/tex]
1. Start with the original inequality:
[tex]\[
-2x + 9 \leq 13
\][/tex]
2. Subtract 9 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
-2x + 9 - 9 \leq 13 - 9
\][/tex]
Simplifying, we get:
[tex]\[
-2x \leq 4
\][/tex]
3. Divide both sides by [tex]\(-2\)[/tex]. Remember, when you divide by a negative number, you must reverse the inequality sign:
[tex]\[
x \geq \frac{4}{-2}
\][/tex]
Simplifying, we get:
[tex]\[
x \geq -2
\][/tex]
### Solving the second inequality: [tex]\(-3x - 1 > 8\)[/tex]
1. Start with the original inequality:
[tex]\[
-3x - 1 > 8
\][/tex]
2. Add 1 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
-3x - 1 + 1 > 8 + 1
\][/tex]
Simplifying, we get:
[tex]\[
-3x > 9
\][/tex]
3. Divide both sides by [tex]\(-3\)[/tex]. Again, remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
x < \frac{9}{-3}
\][/tex]
Simplifying, we get:
[tex]\[
x < -3
\][/tex]
### Combining the solutions
The compound inequality [tex]\( -2x + 9 \leq 13 \)[/tex] or [tex]\( -3x - 1 > 8 \)[/tex] combines our findings from the individual inequalities. The solution to the compound inequality is:
[tex]\[
x \geq -2 \quad \text{or} \quad x < -3
\][/tex]
Therefore, the set of all values of [tex]\(x\)[/tex] which satisfy either inequality is:
[tex]\[
x \geq -2 \quad \text{or} \quad x < -3
\][/tex]
