General Inequalities and Their Applications
Multi-step linear inequalities - Questions
Answers
1. \(r < \frac{4}{7}\)
| \(35 r-21 < -35 r+19\) | |
| \(35 r < -35 r+40\) | Add \(21\) to both sides. |
| \(70 r < 40\) | Add \(35r\) to both sides. |
| \(r < \frac{4}{7}\) | Divide both sides by \(70\) and simplify |
In conclusion, the answer is \(r < \frac{4}{7}\).
2. \(a \leq \frac{39}{5}\)
| \(60 a+64 \geq 80 a-92\) | |
| \(60 a \geq 80 a-156\) | Subtract \(64\) from both sides |
| \(-20 a \geq-156\) | Subtract \(80a\) from both sides |
| \(20 a \leq 156\) | Multiply both sides by \(-1\) |
| \(a \leq \frac{39}{5}\) | Divide both sides by \(20\) and simplify |
Why did the inequality sign flip when we multiplied by \(-1\)?
The inequality sign flips because we order negative numbers differently from positive numbers.
For example, \(2 < 3\). However, when we multiply both sides of the inequality by \(-1\), we see that the inequality flips, because \(-2 < -3\).
In general, if \(u < k\), then it follows that \(-u < -k\).
In conclusion, the answer is \(a \leq \frac{39}{5}\).
3. \(t \leq \frac{12}{23}\)
| \(-48 t+2 \leq-71 t+14\) | |
| \( -48 t \leq-71 t+12\) | Subtract \(2\) from both sides |
| \(23 t \leq 12\) | Add \(71t\) to both sides |
| \(t \leq \frac{12}{23}\) | Divide both sides by \(23\) |
In conclusion, the answer is \(t \leq \frac{12}{23}\).
4. \(w > -1\)
| \(53 w+13 < 56 w+16\) | |
| \(53 w < 56 w+3\) | Subtract \(13\) from both sides |
| \(-3 w < 3\) | Subtract \(56w\) from both sides |
| \(3 w > -3\) | Multiply both sides by \(-1\) |
| \(w > -1\) | Divide both sides by \(3\) and simplify |
Why did the inequality sign flip when we multiplied by -1?
The inequality sign flips because we order negative numbers differently from positive numbers.
For example, \(2 < 3\). However, when we multiply both sides of the inequality by \(-1\), we see that the inequality flips, because \(-2 < -3\).
In general, if \(u < k\), then it follows that \(-u > -k\).
In conclusion, the answer is \(w > -1\).