Linear inequalities in one unknown · Ages 14–15
Compound inequalities: and, or
"And": where they overlap. "Or": both parts together.
How to do it
Method
"And": where they overlap. "Or": both parts together
x > −1 and x ≤ 4: −1 < x ≤ 4
Another way
A number from each stretch
x = 0: 0 > −1 and 0 ≤ 4, both true
In full
Two inequalities joined by "and": x must fit both, so draw both on a number line and take where they overlap. x > −1 and x ≤ 4 is −1 < x ≤ 4. Joined by "or": x need only fit one, so take both parts together. Careful with "or": x < 2 or x < 5 is x < 5 (anything below 5 fits one of them), not x < 2. "And" may have no solution: x > 5 and x < 2 never overlap; "or" may take in every number: x < 5 or x > 2. Solve each inequality on its own first, then combine. Counting whole-number solutions: those of −1 < x ≤ 4 are 0, 1, 2, 3, 4, which is 4 − (−1) = 5 of them. Or test numbers: take one from each stretch of the line and put it into both, checking that "and" makes both true and "or" at least one.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
How many whole-number solutions has −3 < x ≤ 4?
Show the answer and working
Answer: 7
−3 out, 4 in: from the number after −3 up to 4, that is 4 − (−3) = 7 of them.
Example 2
x > 5 and x < 7 has no solution.
True or false?
Show the answer and working
Answer: False
Draw both on a number line: x > 5 runs right, x < 7 runs left. They have solutions only if they overlap, that is if 5 < 7.
Example 3
5x + 6 > 11 and x + 5 ≤ 10: there are whole-number solutions
Show the answer and working
Answer: 4
Solve each first: x > 1, x ≤ 5. They overlap in 1 < x ≤ 5: 5 − (1) = 4 whole numbers.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.x ≥ −1 and x ≤ 8: there are whole-number solutions
Answer:
2.The number line shows the solutions of two inequalities. Joined by "and", what is the solution?
- A. x < −2
- B. −5 < x ≤ −2
- C. −5 ≤ x < −2
- D. x ≥ −5
Choose the right answer
Answer:
3.What is the solution of x < −3 or x < 2?
- A. No solution
- B. x < −3
- C. −3 < x < 2
- D. x < 2
Choose the right answer
Answer:
4.6x + 7 > 1 and x − 3 ≤ 2: there are whole-number solutions
Answer:
5.What is the solution of x ≤ 3 or x > −2?
- A. −2 < x ≤ 3
- B. Every number
- C. x ≤ 3
- D. No solution
Choose the right answer
Answer:
All the answers
- 1. 10
- 2. C. −5 ≤ x < −2
- 3. D. x < 2
- 4. 6
- 5. B. Every number
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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