Inequality word problems describe real-life situations where a quantity must be
greater than, less than, at least, at most, no more than, or no less than another
quantity. The goal is to translate the words into a mathematical inequality, solve
it, and interpret the solution in context.
1. Common inequality words
2. General modeling strategy
Most one-variable application problems follow this process:
- Identify the unknown quantity and assign a variable.
- Write expressions for the important quantities in the problem.
- Choose the inequality symbol from the wording.
- Solve the inequality algebraically.
- Apply real-world restrictions such as nonnegative values or whole numbers.
- Write the final answer as a sentence.
For example, if \(q\) is the number of tickets sold, then \(q\) usually cannot be
negative. If tickets are sold individually, \(q\) should also be a whole number.
3. Grade average problems
A common grade problem asks for the minimum final exam score needed to reach a target
average. If the current average is \(A\), the final exam weight is \(w\), the final
score is \(F\), and the target average is \(T\), the model is
\[
(1-w)A+wF\ge T.
\]
The final score usually has a realistic domain such as
\[
0\le F\le 100.
\]
If the solution requires \(F>100\), the target is not reachable under the usual grading
scale.
4. Profit problems
Profit is revenue minus cost. If each item sells for \(p\), each item costs \(v\) to
make, fixed costs are \(C\), and \(q\) items are sold, then
\[
\text{Profit}=pq-(vq+C)=(p-v)q-C.
\]
To earn at least a target profit \(P\), write
\[
(p-v)q-C\ge P.
\]
If \(q\) represents items, the final answer should usually be rounded up to the next
whole number.
5. Budget problems
Budget problems often ask for the maximum quantity that can be bought. If there is a
fixed fee \(F\), a unit cost \(c\), a quantity \(q\), and a budget \(B\), then the total
cost is
\[
F+cq.
\]
A budget limit gives
\[
F+cq\le B.
\]
If \(q\) must be a whole number, the final answer is usually rounded down to the greatest
valid whole number.
6. Age comparison problems
Age problems often compare future ages. If one person is currently \(A\) years old and
another is \(B\) years old, then after \(t\) years their ages are
\[
A+t
\qquad\text{and}\qquad
B+t.
\]
If the first person must be at least \(k\) times as old as the second person, write
\[
A+t\ge k(B+t).
\]
Since \(t\) means years from now, a normal real-world restriction is
\[
t\ge0.
\]
7. Mixture problems
A mixture problem tracks the amount of pure substance. If \(a\) units of a solution with
concentration \(c_1\) are mixed with \(x\) units of a solution with concentration \(c_2\),
then the new concentration is
\[
\frac{c_1a+c_2x}{a+x}.
\]
To make the mixture at least a target concentration \(c_T\), use
\[
\frac{c_1a+c_2x}{a+x}\ge c_T.
\]
Because \(a+x\) is positive in ordinary mixture problems, multiplying both sides by
\(a+x\) keeps the inequality direction the same.
8. Geometry constraint problems
Geometry word problems often ask for possible dimensions under a perimeter or area
constraint. For a rectangle whose width is \(w\) and length is \(w+d\), the perimeter is
\[
2w+2(w+d).
\]
If the perimeter must be at most \(P\), then
\[
2w+2(w+d)\le P.
\]
Widths and lengths cannot be negative, so the solution must be checked against the
domain \(w\ge0\).
9. Rounding in real-life answers
A numerical inequality solution may be continuous, but the real situation may require
whole numbers. For example:
- If the answer is “at least \(61.54\) items,” you need \(62\) items.
- If the answer is “at most \(10.8\) tickets,” you can buy at most \(10\) tickets.
- If the answer is a test score, decimals may be allowed depending on the grading system.
10. Summary
