A system of inequalities finds the range of x that satisfies two or more inequalities at the same time.
Steps for Solving
Solve each inequality separately, then find the common part of the solutions. Writing them on a number line makes the common range visible.
Example ①
Solve the following system of inequalities.
2x − 1 ≧ 3
x + 4 < 8
Solution:① 2x ≧ 4 → x ≧ 2. ② x < 4.
Common part: 2 ≦ x < 4
Solve the following system of inequalities.
2x − 1 ≧ 3
x + 4 < 8
Solution:① 2x ≧ 4 → x ≧ 2. ② x < 4.
Common part: 2 ≦ x < 4
Form A < B < C
For a compound inequality, solve A < B and B < C separately, then find their common part.
Example ②
Solve 1 < 2x − 3 ≦ 7.
Solution:Adding 3 to each side gives 4 < 2x ≦ 10. Dividing each side by 2 gives 2 < x ≦ 5
Solve 1 < 2x − 3 ≦ 7.
Solution:Adding 3 to each side gives 4 < 2x ≦ 10. Dividing each side by 2 gives 2 < x ≦ 5
Key Points
- Solve each inequality separately and find the common part
- There may be no solution when the common part is empty
- Perform the same operation on each part of a compound inequality
Practice Problems
- Solve 3x − 5 ≦ 7 and 2x + 1 > 3.
- Solve −3 < 5 − 2x ≦ 5.
- Solve 4x − 3 > 5 and 6 − x ≧ 1.
Solutions and Explanations
- Solution:1 < x ≦ 4
Explanation: ① x ≦ 4, ② x > 1. Common part: 1 < x ≦ 4. - Solution:0 ≦ x < 4
Explanation: Subtracting 5 from each part gives −8 < −2x ≦ 0 → dividing each part by −2 and reversing gives 0 ≦ x < 4. - Solution:2 < x ≦ 5
Explanation: ① x > 2, ② x ≦ 5. Common part: 2 < x ≦ 5.