6.5 Solving Linear Inequalities 333 Note that in Example 5, the direction of the inequality symbol did not change when both sides of the inequality were divided by the positive number 3. Thus, the solution set of x3 5 7 − > is all real numbers greater than 4. 0123456789 x . 4 7 Now try Exercise 21 Timely Tip If, while solving an inequality, you get an inequality with no variables that is always • false, such as 3 5 > then the inequality has no solution . • true, such as 2 1, > then the solution is all real numbers. Some linear inequalities have no solution. When solving an inequality, if you obtain a statement with no variables that is always false , such as 3 5, > the inequality has no solution . There is no real number that makes the statement true. Example 6 An Inequality with No Solution Solve the inequality x x x 2( 4) 3 5( 1), − − < − + and graph the solution set on the number line. Solution To solve the inequality, we simplify each side and attempt to isolate x on one side of the inequality. x x x x x x x x x x x x 2( 4) 3 5( 1) 2 8 3 5 5 2 8 2 5 2 8 2 2 5 2 8 5 − − < − + − + < − − − + < − − − + + < − − + < − Distributive property Combined like terms Add x2 to both sides Note the final inequality has no variables. Since it is impossible for 8 to be less than 5, − this inequality is always false and has no solution . The graph of the solution set is an empty number line: 25 24 23 22 21 0 1 2 3 4 5 7 Now try Exercise 29 Some linear inequalities have all real numbers as their solution. When solving an inequality, if you obtain a statement with no variables that is always true , such as 2 1, > the solution is all real numbers. Example 7 An Inequality Whose Solution is All Real Numbers Solve the inequality x x x 3( 4) 2 5( 3), + + > − and graph the solution set on the number line. Solution To solve the inequality, we simplify each side and attempt to isolate x on one side of the inequality. x x x x x x x x x x x x 3( 4) 2 5( 3) 3 12 2 5 15 5 12 5 15 5 12 5 5 15 5 12 15 + + > − + + > − + > − + − > − − > − Distributive property Combined like terms Subtract x5 from both sides Note the final inequality has no variables. Since 12 is always greater than 15, − this inequality is always true. Therefore, the solution is all real numbers . The graph of the solution set is a number line that is shaded everywhere: 25 24 23 22 21 0 1 2 3 4 5 7 Now try Exercise 23
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