Question

Sophie, Adola, and Gilly are debating potential values in the solution set of the compound inequality −2≤2x−2<8 . Sophie says that −2 is a solution. Adola says that 5 is a solution. Gilly says that 2 is a solution. Which person is correct? Enter 1 for Sophie, 2 for Adola, or 3 for Gilly.

Answer 1

To determine which person's claim about a solution is correct, we first need to solve the compound inequality \( -2 \leq 2x - 2 < 8 \).

We can solve it in two parts.

**Part 1: Solve \( -2 \leq 2x - 2 \)**

1. Add 2 to both sides:
\[
-2 + 2 \leq 2x - 2 + 2 \implies 0 \leq 2x
\]
2. Divide by 2:
\[
0 \leq x \implies x \geq 0
\]

**Part 2: Solve \( 2x - 2 < 8 \)**

1. Add 2 to both sides:
\[
2x - 2 + 2 < 8 + 2 \implies 2x < 10
\]
2. Divide by 2:
\[
x < 5
\]

Now, combining both parts, we have:
\[
0 \leq x < 5
\]

This means the solution set consists of all \( x \) values such that \( 0 \leq x < 5 \).

Now, let's check each person's claims:

- **Sophie claims \( -2 \) is a solution:**
\( -2 \) does not satisfy \( x \geq 0 \). Therefore, Sophie's claim is incorrect.

- **Adola claims \( 5 \) is a solution:**
\( 5 \) does not satisfy \( x < 5 \). Therefore, Adola's claim is incorrect.

- **Gilly claims \( 2 \) is a solution:**
\( 2 \) satisfies \( 0 \leq 2 < 5 \). Therefore, Gilly's claim is correct.

Since Gilly is the only one whose value is in the solution set, the answer is:

**3 for Gilly.**