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1.7 and 1.8 Graph an Inequality’s Solution Set Set-builder And Interval Notation LINEAR AND ABSOLUTE VALUE INEQUALITIES Example 1 Graph the solution set of each. a) x3 b) x 1 c) 1 x 3 There are two ways to represent solution sets of inequalities. They are set-builder notation and interval notation. The table below shows the nine possible types of intervals used to describe inequalities. Interval Notation 2, 4 Set-Builder Notation Graph x 2 x 4 x 2 x 4 x 2 x 4 x 2 x 4 x x 2 x x 2 x x 4 x x 4 x x 2, 4 2, 4 2, 4 2, 2, , 4 , 4 , The set of all real numbers Example 2 a) b) c) d) Express the intervals in terms of inequalities and then graph. State whether the interval is bounded or unbounded. 1, 4 0, 9 4, , 5 1 Properties of Inequalities The only thing to remember about solving inequalities is: When multiplying or dividing by a negative number, we must reverse the direction of the inequality. Example 3 Solve. Graph the solution. Write the solution in set-builder notation and interval notation. a) 3x 5 7 b) 4 3x 7 2x c) 3 2x 1 3 Quadratic Inequalities One method of solving a quadratic inequality involves finding the solutions of the corresponding equation, and then testing values in the intervals on a number line determined by these solutions. Steps STEPS FOR SOLVING A QUADRATIC INEQUALITIES Step 1 – Solve the corresponding quadratic equation. Step 2 – Identify the intervals determined by the solutions of the equation. Step 3 – Use a test value from each interval to determine which intervals form the solution set. Example 4 Solve the inequality x 2 x 12 0 Example 5 Solve the inequality 2 2 x 2 5 x 12 0 Rational Inequalities Inequalities involving rational expressions (fractions) are called rational inequalities and are solved in a manner similar to the procedure for solving quadratic inequalities. Steps STEPS FOR SOLVING A RATIONAL INEQUALITIES Step 1 – Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side. Step 2 – Determine the values that will cause either the numerator of the denominator of the rational expression to equal 0. These values determine the intervals on the number line to consider. Step 3 – Use a test value from each interval to determine which intervals form the solution set. 5 1 Example 6 Solve x4 Example 7 Solve 2x 1 3 x 3 3 Solve Inequalities Absolute value means _______________ from 0 on a number line. Involving Absolute Therefore, x 2 means “all the number 2 units away from 0.” Value Graphically, you can think of this: To solve an absolute value inequality, we do the same process, but with Inequalities rather than equalities. x 2 means all the number less than 2 units from 0 on a number line. Graphically, this is: x 2 means all the number more than 2 units from 0 on the number line. Graphically, this is: Example 8 Solve and graph: x4 3 Example 9 Solve and graph: 2x 3 5 4 Extra Practice: Solve each inequality. Write the solution set in interval notation. 1. 6 x 2 x 3 4 x 5 2. 2 4 x 5 x 1 6 x 2 4. 3 2x 7 13 x 1 5 2 6. 1 6x 5 4 7. x 2 x 6 0 8. 2 x 2 9 x 18 0 9. x x 1 6 10. x 2 4 x 4 3. 1 2 1 1 x x x 3 3 5 2 10 5. 4 11. x 3 0 x5 12. x 1 0 x4 13. x6 1 x2 14. 10 5 2x 3 15. 7 1 x2 x2 16. 4 3 x 2 x 1 Review: 17. Solve: 2x 3 x 3 18. Solve by completing the square: x 2 6 x 12 0 5