# 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 InequalitiesLet a, b and c be real numbers.1 - Transitive PropertyIf a < b and b < c then a < c 2 - Addition PropertyIf a < b then a + c < b + c 3 - Subtraction PropertyIf a < b then a - c < b - c 4 - Multiplication Property4 - 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 1Solve the inequalitySolution to Example 1Given 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 ConclusionThe solution set consists of all real numbers in the interval (2 , + ∞). Matched ExerciseSolve the inequality
## Example 2Solve the inequalitySolution to Example 2Given 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
the inequality sign and simplify ( Multiplication Property above)REVERSEx < 4 Conclusion
The solution set consists of all real numbers in the interval (- ∞ , 4)
Matched ExerciseSolve the inequality
## Example 3Solve the double inequality Solution to Example 3Given -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 ConclusionThe solution set consists of all real numbers in the interval (- 2 , 1) Matched ExerciseSolve the double inequality
## Example 4Solve the inequality Solution to Example 4Given (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 ConclusionThe solution set consists of all real numbers in the interval (- ∞ , -23/20). Matched ExerciseSolve the inequality
## Answer to Matched Exercise10x - 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 InequalitiesSolve Equations, Systems of Equations and Inequalities.Tutorial on Inequalities |