# Tutorial on Inequalities

This is a tutorial on solving polynomial and rational inequalities. Examples with step by step detailed solutions are presented

 Example 1: Solve the inequality. 2 - 7 x / 5 > -x + 3 Solution to example 1 multiply both sides by 5 (the LCD) (2 - 7 x / 5)5 > (-x + 3)5 10 - 7 x > -5 x + 15 subtract 10 from both sides. - 7 x > -5 x + 5 add 5 x to both sides. -2 x > 5 divide both sides by -2 and reverse inequality. x < -5 / 2 conclusion: The solution set to the above inequality consists of all real numbers that are less than -5/2. Example 2: Solve the polynomial inequality. -4 x 2 > 4 x - 8 Solution to example 2 Add - 4 x + 8 to both sides to make one side zero -4 x 2 - 4 x + 8 > 0 Divide by the common factor - 4 and reverse the inequality. x 2 + x - 2 < 0 Factor the expression on the left. (x - 1)(x + 2) < 0 make table of signs of x - 1, x + 2 and their product. x -inf -2 1 +inf x - 1 - | - | + x + 2 - | + | + (x - 1)(x + 2) + | - | +  conclusion: The solutions of (x - 1)(x + 2) < 0 are the values of x for which the resulting sign is negative.Thus, the solution set of the given inequality is the interval(-2 , 1). Example 3: Solve the rational inequality. 1 /(x + 4) - 2 / (x - 3) > = 0 Solution to example 3 Add the two rational expressions that are on the left (-x - 11) / [ (x + 4)(x - 3) ] > = 0 make table of signs of -x - 11, x + 4, x - 3 and the whole expression on the left of the inequality. x -inf -11 -4 3 +inf -x - 11 + | - | - | - x - 3 - | - | - | + x + 4 - | - | + | + (-x - 11) / [(x + 4)(x - 3)] + | - | + | -  conclusion: The solutions of 1 /(x + 4) - 2 / (x - 3) > = 0 are the values of x for which (-x - 11) / [(x + 4)(x - 3)] is positive or is equal to zero.Thus, the solution set of the given inequality is the intervals (-inf , -11 ] U (-4 , 3). More references and links on how to Solve Equations, Systems of Equations and Inequalities.