General Inequalities and Their Applications

The approach to solving linear inequalities is similar to equations: first, simplify each side, then isolate a variable by doing the same thing to both sides. Remember to switch the sign when multiplying or dividing by a negative number. This lecture series shows examples of solving inequalities and using them to solve word problems. Watch the videos and complete the interactive exercises.

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