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)

Question

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.