Question

sophia, adola, and Gilly are debating potential values in the solution set of the compound inequality -2 is less than or equal to 2x-2<8. sophie says that -2 is a solution. adola says that 5 is a solution. gilly says 2 is a solution. which person is correct?

Answer 1

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

We can break the compound inequality into two parts:

1. \(-2 \leq 2x - 2\)
2. \(2x - 2 < 8\)

**Solving the first inequality:**

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

Add 2 to both sides:

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

This simplifies to:

\(0 \leq 2x\)

Divide both sides by 2:

\(0 \leq x\)

So, \(x \geq 0\).

**Solving the second inequality:**

\(2x - 2 < 8\)

Add 2 to both sides:

\(2x < 8 + 2\)

This simplifies to:

\(2x < 10\)

Divide both sides by 2:

\(x < 5\).

Now we combine the solutions of the two parts:

The complete solution to the compound inequality is:

\[0 \leq x < 5\]

This means that \(x\) can take values from 0 up to (but not including) 5.

**Evaluating each person's claim:**

1. **Sophie says that -2 is a solution.**
- Since -2 is less than 0, Sophie is incorrect.

2. **Adola says that 5 is a solution.**
- Since 5 is not less than 5, Adola is incorrect.

3. **Gilly says that 2 is a solution.**
- Since 2 is within the range \(0 \leq x < 5\), Gilly is correct.

**Conclusion:**
Only Gilly is correct; the solution set includes the value 2.