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Section 4.3 Solving Compound Inequalities 4.3 Lecture Guide: Solving Compound Inequalities Objective: Identify an inequality that is a contradiction or an unconditional inequality. The algebraic process for solving the inequalities we have examined in the first two sections of this chapter has left a variable term on one side of the inequality. These have all been conditional inequalities. Sometimes the algebraic process for solving an inequality will result in the variable being completely removed from the inequality, which means the inequality is a contradiction or an unconditional inequality. A _____________________ _____________________ is an inequality that is only true for certain values of the variable. An _____________________ _____________________ is an inequality that is true for all values of the variable. A __________________ is an inequality that is not true for any value of the variable. Solve each inequality. Identify each contradiction or unconditional inequality. 1. 3x 2 x 4 2 x Solve each inequality. Identify each contradiction or unconditional inequality. 2. 5x 3 2 x 3x 3 Solve each inequality. Identify each contradiction or unconditional inequality. 3. 4 2 x 3 7 x 12 x Solve each inequality. Identify each contradiction or unconditional inequality. 4. 3 x 4 2 x 5 x 1 Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1. 5. 5x 2 7 x 2 4 x Solution: ____________ Type: __________________ y2 y1 10, 10, 1 by 10, 10, 1 Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1. 6. 2 x x 4 5 x 6 x 1 Solution: ____________ Type: __________________ y2 y1 10, 10, 1 by 10, 10, 1 Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1. 7. 2 x 3 5 x 9 6 x Solution: ____________ Type: __________________ y1 y2 10, 10, 1 by 10, 10, 1 Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1. 8. 3 6 x 5 3x 15 15 x Solution: ____________ Type: __________________ y1 y2 10, 10, 1 by 10, 10, 1 Objective: Solve compound inequalities involving intersection and union. Intersection of Two Sets Algebraic Notation A B Verbally The intersection of A and B is the set that contains the elements in both A and B. Numerical Example 3, 4 0,6 0, 4 Graphical Example Union of Two Sets Algebraic Notation A B Verbally The union of A and B is the set that contains the elements in either A or B or both. Numerical Example 3, 4 0,6 3,6 Graphical Example 9. Complete the following table. Compound Verbal Graph Interval Inequality Description Notation x 3 and x 7 x 2 and x 0 1 x 5 4 x7 x 2 or x 5 x 1 or x 2 10. (a) Using the word ____________ between two inequalities indicates the intersection of two sets. In some cases, an intersection can be written in a combined form that looks like one expression sandwiched between two other expressions. (b) Using the word ____________ between two inequalities indicates the union of two sets. Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B. 11. A (3,6]; B [1, 4] A B = ____________ A B = ____________ Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B. 12. A (3,9); B [2,17) A B = ____________ A B = ____________ Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B. 13. A (,6]; B [1, ) A B = ____________ A B = ____________ Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B. 14. A (, 2); B (5, ) A B = ____________ A B = ____________ Write each inequality as two separate inequalities using the word “and” to connect the inequalities. 15. 0 x2 Write each inequality as two separate inequalities using the word “and” to connect the inequalities. 16. 13 x 3 Write each inequality expression as a single compound inequality. 17. x 2 and x 3 Write each inequality expression as a single compound inequality. 18. x 10 and x 8 Solve each compound inequality. Give the solution in interval notation. 19. 12 6 x 24 Solve each compound inequality. Give the solution in interval notation. 20. 1 2 x 1 3 Solve each compound inequality. Give the solution in interval notation. x 21. 1 3 4 2 Solve each compound inequality. Give the solution in interval notation. 22. 5x 2 6 x 1 5x 4 Solve each compound inequality. Give the solution in interval notation. 23. 3x 2 x 1 or 4 x 1 3 x 2 Solve each compound inequality. Give the solution in interval notation. 24. 3x 2 6 x and 3x 2 12 x Solve each compound inequality. Give the solution in interval notation. x x 25. 2 x 3 5 x 9 or 1 2 3 Solve each compound inequality. Give the solution in interval notation. 26. 5x 1 3x 9 and 4 x 2 7 x 16 27. Use the graph below to determine the solution of x 2 x 6 x 1 5 3 3 3 y 2 y2 x 1 8 3 x y1 6 3 x -8 8 x -8 y3 5 3 Solution: ___________________ 28. The perimeter of the parallelogram shown must be at least 20 cm and no more than 48 cm. If the given length must be 5 cm, determine the possible lengths for x, the unknown dimension. x cm 5 cm 5 cm x cm