These application problems require you to set up an inequality and find the range of the desired quantity.
Key Points for Setting Up Equations
- “At least” and “at most” → include equality (≧, ≦)
- “Greater than” and “less than” → do not include equality (>, <)
- Since quantities and numbers of people are integers, convert the final range to integers
Example ①
You buy a total of 12 snacks at 80 yen each and drinks at 120 yen each. You want the total cost to be no more than 1200 yen. What is the minimum number of snacks you must buy?
Solution:Let x be the number of snacks and (12 − x) the number of drinks.
80x + 120(12 − x) ≦ 1200 → 80x + 1440 − 120x ≦ 1200 → −40x ≦ −240 → x ≧ 6
The minimum is 6.
You buy a total of 12 snacks at 80 yen each and drinks at 120 yen each. You want the total cost to be no more than 1200 yen. What is the minimum number of snacks you must buy?
Solution:Let x be the number of snacks and (12 − x) the number of drinks.
80x + 120(12 − x) ≦ 1200 → 80x + 1440 − 120x ≦ 1200 → −40x ≦ −240 → x ≧ 6
The minimum is 6.
Example ②
When an integer is multiplied by 4 and 7 is added, the result becomes greater than the number obtained by multiplying the integer by 2 and adding 15. Find the smallest such integer.
Solution:4x + 7 > 2x + 15 → 2x > 8 → x > 4. The smallest integer is 5.
When an integer is multiplied by 4 and 7 is added, the result becomes greater than the number obtained by multiplying the integer by 2 and adding 15. Find the smallest such integer.
Solution:4x + 7 > 2x + 15 → 2x > 8 → x > 4. The smallest integer is 5.
Key Points
- Distinguish “at least,” “at most,” “greater than,” and “less than” when converting words into inequality signs
- Convert integer solutions to an integer range at the end
- State clearly what the variable represents at the beginning
Practice Problems
- You buy a total of 10 apples at 120 yen each and oranges at 80 yen each, keeping the total at or below 1000 yen. What is the maximum number of apples you can buy?
- For what range of a is the cost of buying 5 items at a yen each less than 2000 yen?
- When the sum of three consecutive integers is greater than 60, find the minimum value of the smallest integer.
Solutions and Explanations
- Solution:Up to 5
Explanation: Let x be the number of apples and (10−x) the number of oranges. 120x + 80(10−x) ≦ 1000 → 40x ≦ 200 → x ≦ 5. - Solution:a < 400
Explanation: 5a < 2000 → a < 400. - Solution:20
Explanation: x + (x+1) + (x+2) > 60 → 3x + 3 > 60 → x > 19. The smallest integer is 20.