Question

Match each equation with its solution set.
∅
 |5 − x| = 6  |2x + 5| = 9  |x − 3| = −12
 |x + 3| = 12  |2x + 7| + 2 = 11

Answer 1

To match each equation with its solution set, let's analyze each of the equations one by one.

1. **Equation: \(|5 - x| = 6\)**
This can be rewritten as two equations:
\(5 - x = 6\) → \(x = -1\)
\(5 - x = -6\) → \(x = 11\)
**Solution set:** \(\{-1, 11\}\)

2. **Equation: \(|2x + 5| = 9\)**
This can also be rewritten as two equations:
\(2x + 5 = 9\) → \(2x = 4\) → \(x = 2\)
\(2x + 5 = -9\) → \(2x = -14\) → \(x = -7\)
**Solution set:** \(\{-7, 2\}\)

3. **Equation: \(|x - 3| = -12\)**
An absolute value cannot equal a negative number.
**Solution set:** \(\emptyset\)

4. **Equation: \(|x + 3| = 12\)**
Rewrite as two equations:
\(x + 3 = 12\) → \(x = 9\)
\(x + 3 = -12\) → \(x = -15\)
**Solution set:** \(\{-15, 9\}\)

5. **Equation: \(|2x + 7| + 2 = 11\)**
First, isolate the absolute value:
\(|2x + 7| = 9\)
Rewrite as:
\(2x + 7 = 9\) → \(2x = 2\) → \(x = 1\)
\(2x + 7 = -9\) → \(2x = -16\) → \(x = -8\)
**Solution set:** \(\{-8, 1\}\)

Now we summarize the solution sets:

- \(|5 - x| = 6\) has the solution set \(\{-1, 11\}\)
- \(|2x + 5| = 9\) has the solution set \(\{-7, 2\}\)
- \(|x - 3| = -12\) has the solution set \(\emptyset\)
- \(|x + 3| = 12\) has the solution set \(\{-15, 9\}\)
- \(|2x + 7| + 2 = 11\) has the solution set \(\{-8, 1\}\)

### Final match:
- ∅: \(|x - 3| = -12\)
- \(\{-1, 11\}\): \(|5 - x| = 6\)
- \(\{-7, 2\}\): \(|2x + 5| = 9\)
- \(\{-15, 9\}\): \(|x + 3| = 12\)
- \(\{-8, 1\}\): \(|2x + 7| + 2 = 11\)