The common solution set of the inequations `5 le 2x + 7 le 8 and 7 le 3x + 5 le 9` is _____

Options

A. `(2)/(3) le x le (4)/(3)`
B. `-1 le x le (4)/(3)`
C. `(2)/(3) le x le (1)/(2)`
D. Null set

Explanation

To find the common solution set of the inequations \(5 \leq 2x + 7 \leq 8\) and \(7 \leq 3x + 5 \leq 9\), we will solve each part step by step.

### Step 1: Solve the first inequation \(5 \leq 2x + 7 \leq 8\)

1. **Split the compound inequality** into two separate inequalities:
- \(5 \leq 2x + 7\)
- \(2x + 7 \leq 8\)

2. **Solve the first part** \(5 \leq 2x + 7\):
- Subtract 7 from both sides:
\[ 5 - 7 \leq 2x \implies -2 \leq 2x \]
- Divide by 2:
\[ -1 \leq x \implies x \geq -1 \]

3. **Solve the second part** \(2x + 7 \leq 8\):
- Subtract 7 from both sides:
\[ 2x \leq 8 - 7 \implies 2x \leq 1 \]
- Divide by 2:
\[ x \leq \frac{1}{2} \]

4. **Combine the results** from the first inequation:
\[ -1 \leq x \leq \frac{1}{2} \]

### Step 2: Solve the second inequation \(7 \leq 3x + 5 \leq 9\)

1. **Split the compound inequality** into two separate inequalities:
- \(7 \leq 3x + 5\)
- \(3x + 5 \leq 9\)

2. **Solve the first part** \(7 \leq 3x + 5\):
- Subtract 5 from both sides:
\[ 7 - 5 \leq 3x \implies 2 \leq 3x \]
- Divide by 3:
\[ \frac{2}{3} \leq x \implies x \geq \frac{2}{3} \]

3. **Solve the second part** \(3x + 5 \leq 9\):
- Subtract 5 from both sides:
\[ 3x \leq 9 - 5 \implies 3x \leq 4 \]
- Divide by 3:
\[ x \leq \frac{4}{3} \]

4. **Combine the results** from the second inequation:
\[ \frac{2}{3} \leq x \leq \frac{4}{3} \]

### Step 3: Find the common solution set

Now we have two ranges:
1. From the first inequation: \(-1 \leq x \leq \frac{1}{2}\)
2. From the second inequation: \(\frac{2}{3} \leq x \leq \frac{4}{3}\)

To find the common solution, we look for the intersection of these two ranges.

- The first range ends at \(\frac{1}{2}\), which is less than \(\frac{2}{3}\).
- Therefore, there is no overlap between the two ranges.

### Conclusion

The common solution set of the inequations \(5 \leq 2x + 7 \leq 8\) and \(7 \leq 3x + 5 \leq 9\) is **empty**.