Question

For which of the following compound inequalities is there no solution?
a)3m+12>33\ and\ -4m\ge 32
b)5m>15\ and\ -6m<18
c)-2m-12\le -24\ and\ 7m+8\ge 22
d)-3m\ge 12\ and\ m-6<3

Answer 1

To determine which compound inequalities have no solution, let’s solve each one step by step.

**a)** \(3m + 12 > 33\) and \(-4m \ge 32\)

1. Solve \(3m + 12 > 33\):
\[
3m > 33 - 12 \\
3m > 21 \\
m > 7
\]

2. Solve \(-4m \ge 32\):
\[
m \le -8
\]

Now we have:
- \(m > 7\)
- \(m \le -8\)

These two conditions cannot be true at the same time. So, **this compound inequality has no solution**.

---

**b)** \(5m > 15\) and \(-6m < 18\)

1. Solve \(5m > 15\):
\[
m > 3
\]

2. Solve \(-6m < 18\):
\[
m > -3
\]

Now we have:
- \(m > 3\)
- \(m > -3\)

Since \(m > 3\) is stronger than \(m > -3\), the solution is \(m > 3\) which is valid.

---

**c)** \(-2m - 12 \le -24\) and \(7m + 8 \ge 22\)

1. Solve \(-2m - 12 \le -24\):
\[
-2m \le -24 + 12 \\
-2m \le -12 \\
m \ge 6
\]

2. Solve \(7m + 8 \ge 22\):
\[
7m \ge 22 - 8 \\
7m \ge 14 \\
m \ge 2
\]

Now we have:
- \(m \ge 6\)
- \(m \ge 2\)

The more restrictive condition is \(m \ge 6\), which is valid.

---

**d)** \(-3m \ge 12\) and \(m - 6 < 3\)

1. Solve \(-3m \ge 12\):
\[
m \le -4
\]

2. Solve \(m - 6 < 3\):
\[
m < 9
\]

Now we have:
- \(m \le -4\)
- \(m < 9\)

Both inequalities can hold together. The solution set is \(m \le -4\).

---

### Conclusion
The compound inequality with **no solution** is:

**a)** \(3m + 12 > 33\) and \(-4m \ge 32\).