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

Question

. 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.