Linear inequalities in one unknown · Ages 14–15
Word problems with inequalities
At least ≥, at most ≤; at most rounds down, at least rounds up.
How to do it
Method
At least ≥, at most ≤; at most rounds down, at least rounds up
6x ≤ 50: x ≤ 8.33…, at most 8 pens
Another way
Find the boundary, then check
6 × 8 = 48 ≤ 50, 6 × 9 = 54 > 50
In full
Choose the unknown, then find the words about size: "at least" and "not less than" are ≥; "at most", "not more than" and "not over" are ≤; "more than" and "over" are >; "less than" is <. Form the inequality, solve it, and take a whole number the way the story needs. $50 buys pens at $6 each: at most how many? 6x ≤ 50, x ≤ 8.33…, and a count is whole, so at most 8 (round down). Saving at least $200 at $30 a week: at least how many weeks? 30x ≥ 200, x ≥ 6.66…, so at least 7 weeks (round up). Or find the boundary first: the equation 6x = 50 gives 8.33…; then see which way to go to a whole number, and check the answer in the story: 6 × 8 = 48 ≤ 50, while 6 × 9 = 54 is too much.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A lift's sign says "no more than 17 passengers". With x passengers, how is that written as an inequality?
- A. x ≤ 17
- B. x ≥ 17
- C. x > 17
- D. x < 17
Choose the right answer
Show the answer and working
Answer: A. x ≤ 17
"No more than 17" is at most 17: x ≤ 17.
Example 2
4 times a number, take away 5, is not less than 37. What is the least whole number it can be?
Show the answer and working
Answer: 11
With the number x: 4x − 5 ≥ 37, 4x ≥ 42, x ≥ 424. The least whole number from there is 11.
Example 3
A gym charges two ways: plan A, $67 a visit; plan B, $260 a month plus $29 a visit. At least how many visits a month make plan B cheaper than A?
Show the answer and working
Answer: 7 visits
With x visits: 260 + 29x < 67x, 260 < 38x, x > 26038. The least whole number above that is 7.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A lift's sign says "no more than 22 passengers". With x passengers, how is that written as an inequality?
- A. x ≥ 22
- B. x < 22
- C. x ≤ 22
- D. x > 22
Choose the right answer
Answer:
2.With $89, at most how many pens at $9 each can be bought?
Answer:
3.A lift carries at most 630 kg. A 58 kg delivery worker rides with boxes of 30 kg each. At most how many boxes can go in?
Answer:
4.A rectangular flower bed is 12 m longer than it is wide, and is to be fenced with no more than 54 m of fencing. If its width is a whole number of metres, at most how wide can it be?
Answer:
5.A gym charges two ways: plan A, $64 a visit; plan B, $200 a month plus $26 a visit. At least how many visits a month make plan B cheaper than A?
Answer:
All the answers
- 1. C. x ≤ 22
- 2. 9 pens
- 3. 19 boxes
- 4. 7 m
- 5. 6 visits
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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