Question

show the steps to solve
−88 ≤−2(4+5x)≤62

Answer 1

To solve the compound inequality \(-88 \leq -2(4 + 5x) \leq 62\), we will break it down into two parts and solve each inequality step by step.

---

### Step 1: Solve the left part of the compound inequality

1. **Start with**:
\[
-88 \leq -2(4 + 5x)
\]

2. **Distribute**:
\[
-88 \leq -8 - 10x
\]

3. **Isolate the term involving \(x\)**:
- Add \(8\) to both sides:
\[
-80 \leq -10x
\]

4. **Divide by \(-10\)**: (Remember to reverse the inequality sign when dividing by a negative number)
\[
8 \geq x
\]
or
\[
x \leq 8
\]

---

### Step 2: Solve the right part of the compound inequality

1. **Start with**:
\[
-2(4 + 5x) \leq 62
\]

2. **Distribute**:
\[
-8 - 10x \leq 62
\]

3. **Isolate the term involving \(x\)**:
- Add \(8\) to both sides:
\[
-10x \leq 70
\]

4. **Divide by \(-10\)**: (Again, reverse the inequality sign)
\[
x \geq -7
\]

---

### Step 3: Combine the results from both parts

From the left part, we found \(x \leq 8\).
From the right part, we found \(x \geq -7\).

Thus, the solution to the compound inequality is:
\[
-7 \leq x \leq 8
\]

### Final Answer
In interval notation, the solution can be written as:
\[
[-7, 8]
\]