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.(1 point)
The person who is correct is person .

Answer 1

To find out who is correct, we need to solve the compound inequality \( -2 \leq 2x - 2 < 8 \).

1. **Break the compound inequality into two parts:**

\(-2 \leq 2x - 2\)

and

\(2x - 2 < 8\).

2. **Solve the first part:**

\(-2 \leq 2x - 2\)

Add 2 to both sides:

\(-2 + 2 \leq 2x\)

\(0 \leq 2x\)

Divide both sides by 2:

\(0 \leq x\) or \(x \geq 0\).

3. **Solve the second part:**

\(2x - 2 < 8\)

Add 2 to both sides:

\(2x < 8 + 2\)

\(2x < 10\)

Divide both sides by 2:

\(x < 5\).

4. **Combine the results:**

The solution to the compound inequality is:

\(0 \leq x < 5\)

This means \(x\) can be any value starting from \(0\) up to but not including \(5\).

5. **Evaluate each person's claim:**

- **Sophie claims \( -2 \) is a solution:**
- \( -2 \) is **not** within the interval \( [0, 5) \).

- **Adola claims \( 5 \) is a solution:**
- \( 5 \) is also **not** within the interval \( [0, 5) \) (it is at the boundary and not included).

- **Gilly claims \( 2 \) is a solution:**
- \( 2 \) is **within** the interval \( [0, 5) \).

So, Gilly is the only one who is correct.

The person who is correct is person **3**.