Section 3.8: Real-Life Applications
Algebraic inequalities and absolute value concepts can model real-world situations such as budgeting, distances, and tolerances.
Example 1: Budget Problem
A student has $50 to spend on books. Each book costs $8. Write and solve an inequality to determine the maximum number of books the student can buy.
Let \( x \) = number of books.
Inequality: \( 8x \le 50 \)
Solution: \( x \le 6.25 \Rightarrow \) maximum of 6 books.
Example 2: Distance Tolerance
A machine part must be within 0.02 cm of 5 cm. Write and solve an absolute value inequality for acceptable lengths.
Inequality: \( |x - 5| \le 0.02 \)
Solution: \( 4.98 \le x \le 5.02 \)
Practice Problems
- A taxi charges a $3 flat fee plus $2 per mile. Write an inequality for trips costing less than $15.
- A recipe allows 0.5 ± 0.05 cups of sugar. Write an absolute value inequality for acceptable sugar amounts.
- A student must score at least 70 on a test. Write and solve an inequality for passing scores.
- The temperature in a greenhouse should remain within 22 ± 2°C. Write an absolute value inequality for acceptable temperatures.
- A car’s speed limit is 60 mph. Write an inequality for legal speeds.