To determine who is correct in their assessment of the solutions for the compound inequality \(-2 \leq 2x - 2 < 8\), we need to solve the inequality step by step.
Step 1: Break down the compound inequality
The compound inequality consists of two parts:
- \(-2 \leq 2x - 2\)
- \(2x - 2 < 8\)
Step 2: Solve the first part
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\(-2 \leq 2x - 2\)
To isolate \(x\), we'll first add 2 to both sides: \[ -2 + 2 \leq 2x - 2 + 2 \implies 0 \leq 2x \] Now divide both sides by 2: \[ 0 \leq x \implies x \geq 0 \]
Step 3: Solve the second part
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\(2x - 2 < 8\)
To isolate \(x\), we first add 2 to both sides: \[ 2x - 2 + 2 < 8 + 2 \implies 2x < 10 \] Now divide both sides by 2: \[ x < 5 \]
Step 4: Combine the results
The solution set for the compound inequality \(-2 \leq 2x - 2 < 8\) is: \[ 0 \leq x < 5 \]
This means that any \(x\) that falls within the interval \([0, 5)\) is a solution to the original inequality.
Step 5: Check each person's claim
- Sophie claims \(x = -2\): This is not a solution because \(-2 < 0\).
- Adola claims \(x = 5\): This is not a solution because \(5\) is not less than \(5\).
- Gilly claims \(x = 2\): This is a solution because \(2\) is within the interval \([0, 5)\).
Conclusion
Only Gilly is correct. Therefore, the answer is:
3