Linear inequalities are solved by transposing terms in the same way as equations, but be careful: when multiplying or dividing by a negative number, the direction of the inequality sign is reversed.
Properties of Inequalities
- Adding or subtracting the same number from both sides does not change the direction
- Multiplying or dividing both sides by a positive number does not change the direction
- Multiplying or dividing both sides by a negative number reverses the direction
Example ①
Solve 3x − 7 < 5.
Solution:3x < 12 → x < 4
Solve 3x − 7 < 5.
Solution:3x < 12 → x < 4
Example ②
Solve −2x + 5 ≧ 11.
Solution:−2x ≧ 6. Dividing both sides by −2 reverses the inequality sign: x ≦ −3
Solve −2x + 5 ≧ 11.
Solution:−2x ≧ 6. Dividing both sides by −2 reverses the inequality sign: x ≦ −3
Inequalities Containing Fractions and Decimals
It is easier to work with them by multiplying both sides by the same positive number to clear the denominators.
Example ③
Solve (x − 1)/2 + (x + 1)/3 > 1.
Solution:Multiply both sides by 6: 3(x−1) + 2(x+1) > 6 → 3x − 3 + 2x + 2 > 6 → 5x > 7 → x > 7/5
Solve (x − 1)/2 + (x + 1)/3 > 1.
Solution:Multiply both sides by 6: 3(x−1) + 2(x+1) > 6 → 3x − 3 + 2x + 2 > 6 → 5x > 7 → x > 7/5
Key Points
- Multiplying or dividing by a negative number → reverse the inequality sign
- Solutions can be shown as a range on a number line
- For fractions, multiply both sides by the least common multiple of the denominators to clear them
Practice Problems
- Solve 4x + 3 ≧ 11.
- Solve −3x + 5 < 14.
- Solve (2x − 1)/3 ≦ x + 1.
Solutions and Explanations
- Solution:x ≧ 2
Explanation: 4x ≧ 8, so x ≧ 2. - Solution:x > −3
Explanation: −3x < 9; dividing both sides by −3 gives x > −3 (reversed). - Solution:x ≧ −4
Explanation: Multiply both sides by 3: 2x − 1 ≦ 3x + 3 → −x ≦ 4 → x ≧ −4.