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 \(w > -1\).