Let's solve the compound inequality step-by-step.
The given inequality is:
[tex]\[
-8 \leq \frac{-3x + 1}{4} \leq 2
\][/tex]
This is a compound inequality composed of two simpler inequalities:
1. [tex]\(-8 \leq \frac{-3x + 1}{4}\)[/tex]
2. [tex]\(\frac{-3x + 1}{4} \leq 2\)[/tex]
We will solve each part separately and then find the intersection of the solutions.
### Solving [tex]\(-8 \leq \frac{-3x + 1}{4}\)[/tex]
Step 1: Eliminate the fraction by multiplying all terms by 4.
[tex]\[
4 \cdot (-8) \leq -3x + 1 \cdot 4
\][/tex]
[tex]\[
-32 \leq -3x + 1
\][/tex]
Step 2: Isolate the term with [tex]\(x\)[/tex] by subtracting 1 from both sides.
[tex]\[
-32 - 1 \leq -3x
\][/tex]
[tex]\[
-33 \leq -3x
\][/tex]
Step 3: Solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-3\)[/tex]. Remember, when you divide by a negative number, the inequality sign reverses.
[tex]\[
\frac{-33}{-3} \geq x
\][/tex]
[tex]\[
11 \geq x
\][/tex]
or
[tex]\[
x \leq 11
\][/tex]
### Solving [tex]\(\frac{-3x + 1}{4} \leq 2\)[/tex]
Step 1: Eliminate the fraction by multiplying all terms by 4.
[tex]\[
\frac{-3x + 1}{4} \cdot 4 \leq 2 \cdot 4
\][/tex]
[tex]\[
-3x + 1 \leq 8
\][/tex]
Step 2: Isolate the term with [tex]\(x\)[/tex] by subtracting 1 from both sides.
[tex]\[
-3x + 1 - 1 \leq 8 - 1
\][/tex]
[tex]\[
-3x \leq 7
\][/tex]
Step 3: Solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-3\)[/tex]. Remember, when you divide by a negative number, the inequality sign reverses.
[tex]\[
\frac{7}{-3} \geq x
\][/tex]
[tex]\[
-\frac{7}{3} \geq x
\][/tex]
or
[tex]\[
x \leq -\frac{7}{3}
\][/tex]
### Combining the solutions
We have two conditions:
1. [tex]\(x \leq 11\)[/tex]
2. [tex]\(x \leq -\frac{7}{3}\)[/tex]
The solution to the compound inequality will be the intersection of these two solutions. The intersection of [tex]\(x \leq 11\)[/tex] and [tex]\(x \leq -\frac{7}{3}\)[/tex] is:
[tex]\[
x \leq -\frac{7}{3}
\][/tex]
Therefore, the solution to the compound inequality is:
[tex]\[
x \leq -\frac{7}{3}
\][/tex]
