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 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 1Solve the inequalityGiven 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 4i above) x > 2 Conclusion The solution set consists of all real numbers in the interval (2 , + ?). Matched Exercise Solve the inequality
Example 2Solve the inequalitySolution 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
Example 3Solve the double inequality 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
Example 4Solve the inequality 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
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 