Integrated Math 1 / Unit 2: Equations and Inequalities
2.6 Modeling with Inequalities
Notes
Use an inequality when a situation has a limit. Phrases like "at most" ($\leq$), "at least" ($\geq$), "fewer than" ($<$) and "more than" ($>$) show which symbol to use.
- Write the inequality, solve it, then decide what answers make sense. Counting numbers must be whole numbers.
- Example: you have $40; a ticket costs $9 and snacks cost $2.50 each. $9 + 2.5s \leq 40 \Rightarrow s \leq 12.4$, so at most 12 snacks.
- Always round in the direction that keeps you within the limit.
Practice
You have $40. A ticket costs $9 and each snack costs $2.50. What is the greatest number of snacks you can buy?
$2.5s \leq 31$, so $s \leq 12.4$. You cannot buy part of a snack, so 12.
A scooter can carry at most 220 pounds. A rider weighs 150 pounds and carries a bag weighing $w$ pounds. Which inequality shows the allowed bag weight?
Total weight must be at most 220, and it is $150 + w$. So $w \leq 70$.
Solve $\displaystyle -\frac{x}{4} \geq 3$.
Multiply by $-4$ and reverse the sign: $x \leq -12$.
Ready for more? Take the Unit 2 test.