Inequality Word Problems
Model real-world constraints as inequalities and solve. Cover budget, minimum/maximum, and range problems. Practice interpreting solutions within the context of the original scenario.
A lesson of Algebra 1, in Linear Inequalities & Absolute Value, about 35 minutes.
By the end of this lesson, you will be able to turn a constraint written in words into an inequality, solve it, and interpret the answer sensibly, including deciding which way to round when the quantity has to be a whole number.
Constraints Are Inequalities
A budget is not an equation. Nobody spends exactly their whole budget on purpose, and nobody is told to score exactly $80$.
Spend at most $$120$. Score at least $80$. Be between $48$ and $72$ inches tall.
Each of those describes a range of acceptable values, which is what an inequality is for. The setup is the same protocol as any word problem, with one extra decision at the start: which of the four signs the phrase demands.
| The phrase | The sign |
|---|---|
| at most, no more than, a maximum of, cannot exceed | $\leq$ |
| at least, no less than, a minimum of | $\geq$ |
| fewer than, less than, under | $<$ |
| more than, greater than, over | $>$ |
| between $a$ and $b$, inclusive | $a \leq x \leq b$ |
Two of those are strict and two are not, and the difference is whether the boundary itself is allowed. Spending exactly $$120$ of a $$120$ budget is fine, so "at most" includes its endpoint. Scoring "more than $80$" does not include $80$.
The Part That Is Not Algebra
An inequality's answer is a range, and real quantities often cannot take every value in it.
$$f \leq 9.375$$
If $f$ counts food items, then $9.375$ items is not a thing. The honest answer is $9$, and the rounding went down even though $0.375$ would normally round up, because $10$ items breaks the constraint.
$$w > 171.4$$
If $w$ counts widgets, the answer is $172$, rounding up, because $171$ does not clear the bar.
Round toward the side that keeps the constraint true. For "at most", round down. For "at least", round up.
This is the step that gets missed, and it is the one that decides whether the answer is useful to anybody.