To determine which compound inequalities have no solution, let’s solve each one step by step.
a) \(3m + 12 > 33\) and \(-4m \ge 32\)
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Solve \(3m + 12 > 33\): \[ 3m > 33 - 12 \ 3m > 21 \ m > 7 \]
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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 \(-6m < 18\)
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Solve \(5m > 15\): \[ m > 3 \]
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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 \(7m + 8 \ge 22\)
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Solve \(-2m - 12 \le -24\): \[ -2m \le -24 + 12 \ -2m \le -12 \ m \ge 6 \]
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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 \(m - 6 < 3\)
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Solve \(-3m \ge 12\): \[ m \le -4 \]
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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 \(-4m \ge 32\).
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