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 of the proposed values are solutions to the compound inequality \( -2 \leq 2x - 2 < 8 \), we first solve the inequality.

### Step 1: Break it down into two parts

1. **Solve the first part:**
\(-2 \leq 2x - 2\)

Add 2 to both sides:
\(-2 + 2 \leq 2x - 2 + 2\)
\(0 \leq 2x\)
Divide by 2:
\(0 \leq x\)
which means \(x \geq 0\).

2. **Solve the second part:**
\(2x - 2 < 8\)

Add 2 to both sides:
\(2x - 2 + 2 < 8 + 2\)
\(2x < 10\)
Divide by 2:
\(x < 5\).

### Step 2: Combine the inequalities

From the two parts, we have:
\[ 0 \leq x < 5 \]
or equivalently,
\[ x \in [0, 5) \]

### Step 3: Check the proposed solutions

1. **Sophie says that \(-2\) is a solution.**
Check if \(-2\) is within the interval \([0, 5)\):
\(-2\) is **not** in \([0, 5)\).

2. **Adola says that \(5\) is a solution.**
Check if \(5\) is within the interval \([0, 5)\):
\(5\) is **not** included in the interval because of the strict inequality.

3. **Gilly says that \(2\) is a solution.**
Check if \(2\) is within the interval \([0, 5)\):
\(2\) is indeed in the interval \([0, 5)\).

### Conclusion

The only correct solution among the three is provided by Gilly. Therefore, the answer is:

**3 for Gilly.**