# Solve Linear Inequalities - Tutorial

Solve linear inequalities: A tutorial with examples and detailed solutions. Double inequalities and inequalities with fractional expressions are also included.
We, first, review some of the properties of the inequalities.

## Properties of Inequalities

Let a, b and c be real numbers.
1 - Transitive Property
If a < b and b < c then a < c
If a < b then a + c < b + c
3 - Subtraction Property
If a < b then a - c < b - c
4 - Multiplication Property
4 - i ) If a < b and c is positive then c a < c b
4 - ii) If a < b and c is negative c a > c b
Note:
If each inequality sign is reversed in the above properties, we obtain similar properties.
If the inequality sign < is replaced by ≤ ( less than or equal) or the sign > is replaced by ≥ ( greater than or equal ), we also obtain similar properties.

## Example 1

Solve the inequality
6x - 6 > 2x + 2
Solution to Example 1
Given
6x - 6 > 2x + 2
Add 6 to both sides and simplify (Property 2 above)
6x > 2x + 8
Subtract 2x to both sides and simplify (Property 3 above)
4x > 8
Multiply both sides by 1/4; and simplify ( Property 4-i above)
x > 2
Conclusion
The solution set consists of all real numbers in the interval (2 , + ∞).
Matched Exercise
Solve the inequality
10x - 8 > 4x + 10

### Example 2

Solve the inequality
2(3x + 2) - 20 > 8(x - 3)

Solution to Example 2
Given
2(3x + 2) -20 > 8(x - 3)
Multiply factors and group like terms
6x + 4 -20 > 8x - 24
6x - 16 > 8x - 24
Add 16 to both sides and simplify (Property 2 above)
6x > 8x - 8
Subtract 8x to both sides and simplify ( Property 3 above)
-2x > -8
Multiply both sides by -1/2 and REVERSE the inequality sign and simplify ( Multiplication Property above)
x < 4
Conclusion The solution set consists of all real numbers in the interval (- ∞ , 4)
Matched Exercise
Solve the inequality
-3(4x + 1) + 10 > -4(x - 3)

### Example 3

Solve the double inequality
-3 < 4(x + 2) - 3 < 9

Solution to Example 3
Given
-3 < 4(x + 2) - 3 < 9
Multiply factors and group like terms
-3 < 4 x + 8 - 3 < 9
-3 < 4 x + 5 < 9
Subtract 5 to all three terms and simplify
-3 - 5 < 4 x + 5 - 5 < 9 - 5
-8 < 4 x < 4
Divide all three terms by 4
-2 < x < 1
Conclusion
The solution set consists of all real numbers in the interval (- 2 , 1)

Matched Exercise
Solve the double inequality
-1 < -2(x - 3) - 3 < 7

### Example 4

Solve the inequality
(x + 2) / 3 - 2 / 5 < (- x - 1) / 3 - 1 / 6

Solution to Example 4
Given
(x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6
Multiply all terms by 30, the LCD
30 (x + 2) /3 - 30 * 2 / 5 < 30(-x - 1) / 3 - 30 * 1 / 6
simplify
10(x + 2) - 6*2 < 10(-x - 1) - 5
Multiply factors and group like terms
10 x + 20 - 12 < - 10 x - 10 - 5
10 x + 8 < - 10 x -15
Subtract 8 to both sides and simplify
10 x < - 10 x - 23
Add 10x to both sides and simplify
20 x < - 23
Divide both sides by 20
x < -23 / 20
Conclusion
The solution set consists of all real numbers in the interval
(- ∞ , -23/20).
Matched Exercise
Solve the inequality
(x - 2) / 4 - 2 / 7 < (-x + 3) / 7 - 1 / 2

10x - 8 > 4x + 10
Ans: x > 3

-3(4x + 1) + 10 > -4(x - 3)
Ans: x < - 5 / 8

-1 < -2(x - 3) - 3 < 7
Ans: -2 < x < 2

(x - 2) / 4 - 2 / 7 < (-x + 3) / 7 - 1 / 2
Ans: x < 20 / 11

## More References and Links to Inequalities

Solve Equations, Systems of Equations and Inequalities.
Tutorial on Inequalities