Inequality Word Problems

In the real world, we rarely deal with exact amounts. Instead, we deal with limits, budgets, and minimum requirements. To solve these situations mathematically, we translate real-world constraints into inequalities.

Translating Words to Math

The most important step in solving an inequality word problem is figuring out which inequality symbol to use. Here is a quick guide to common phrases:

• Greater than ($>$): "more than", "over", "exceeds"
• Less than ($<$): "fewer than", "under", "less than"
• Greater than or equal to ($\geq$): "at least", "no less than", "minimum"
• Less than or equal to ($\leq$): "at most", "no more than", "maximum"

Solving Real-World Examples

Let's look at how to set up and solve these problems step-by-step.

Example 1: The Book Budget

Problem: Books cost \5$each. How many can you buy with at most$$30$?

1. Define the variable: Let $b$ be the number of books.
2. Translate the constraint: The total cost ($5b$) must be "at most" \30$. "At most" means less than or equal to. $5b \leq 30$
3. Solve the inequality: Divide both sides by $5$. $b \leq 6$
4. Check the context: You can buy $6$ books or fewer. Since you cannot buy a negative number of books, the sensible answer is any whole number from $0$ to $6$.

Example 2: The Elevator Weight Limit

Problem: An elevator holds at most $800\text{ kg}$. Five people weigh $350\text{ kg}$ total. How much more weight can be added?

1. Define the variable: Let $w$ be the additional weight.
2. Translate the constraint: The current weight plus the new weight must not exceed the maximum capacity. $350 + w \leq 800$
3. Solve the inequality: Subtract $350$ from both sides. $w \leq 450$
4. Check the context: You can safely add up to $450\text{ kg}$ of additional weight to the elevator.

Always Check the Context

When you finish solving an inequality word problem, always ask yourself: "Does this answer make sense in real life?"

• If you are solving for a number of people, animals, or solid items (like books), your answer must be a whole number.
• If you are solving for time, weight, or distance, decimals and fractions are perfectly fine!
• Most real-world quantities cannot be negative, so an answer like $x \leq 10$ usually implies $0 \leq x \leq 10$ in reality.