Question

Solve the compound inequality.
6 - 2(4x + 1) /geq -10 and 5-(x-9) < 2x +

Answer 1

To solve the compound inequality, we will break it into two separate inequalities and solve each one.

### First Inequality:
\[
6 - 2(4x + 1) \geq -10
\]

1. Distribute \( -2 \):
\[
6 - 8x - 2 \geq -10
\]

2. Combine like terms:
\[
4 - 8x \geq -10
\]

3. Subtract 4 from both sides:
\[
-8x \geq -14
\]

4. Divide both sides by \(-8\) (and remember to flip the inequality sign):
\[
x \leq \frac{14}{8}
\]

5. Simplify:
\[
x \leq \frac{7}{4}
\]

### Second Inequality:
\[
5 - (x - 9) < 2x
\]

1. Distribute the negative sign:
\[
5 - x + 9 < 2x
\]

2. Combine like terms:
\[
14 - x < 2x
\]

3. Add \(x\) to both sides:
\[
14 < 3x
\]

4. Divide both sides by 3:
\[
\frac{14}{3} < x
\]

This can also be written as:
\[
x > \frac{14}{3}
\]

### Final Solution:
Now we combine the two results we found:
1. From the first inequality: \( x \leq \frac{7}{4} \)
2. From the second inequality: \( x > \frac{14}{3} \)

### Conclusion:
The two conditions \( x \leq \frac{7}{4} \) and \( x > \frac{14}{3} \) do not overlap, since \( \frac{7}{4} = 1.75 \) and \( \frac{14}{3} \approx 4.67 \). As a result, there is no solution that satisfies both inequalities simultaneously.

Thus, the solution set is **empty**.