Question

Sophie, Adola, and Gilly are debating potential values in the solution set of the compound inequality -2<2x-2<8.

Answer 1

To solve the compound inequality \(-2 < 2x - 2 < 8\), we can break it down into two parts and solve each part separately.

1. **Solve the left part of the inequality:**
\[
-2 < 2x - 2
\]
Add 2 to both sides:
\[
0 < 2x
\]
Divide by 2:
\[
0 < x \quad \text{or} \quad x > 0
\]

2. **Solve the right part of the inequality:**
\[
2x - 2 < 8
\]
Add 2 to both sides:
\[
2x < 10
\]
Divide by 2:
\[
x < 5
\]

Now, we can combine the two results:
\[
0 < x < 5
\]

Thus, the solution set for the compound inequality \(-2 < 2x - 2 < 8\) is:
\[
(0, 5)
\]

This means that the potential values for \(x\) lie in the interval greater than 0 and less than 5.