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
2 - Addition Property
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

Answer to Matched Exercise


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
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