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Warm-up – pick up handout up front 1. Solve by factoring. Answers: 1000x3-10x 1.x=0, x=1/10, x= -1/10 Solve for x 2. 2. 3x 2 4 4 HW 1.7A (2-14 evens, 21-24, 39-47) 2 3 HW 1.6B (31-39 all) HW 1.6C (61-75 odds) 31.x 4 32.x 9 33. x 13 34. x -1 35.x 5 4 36.x 5 27 37. x 68 38. x 3 39.x 5and 4 61.x 8and 8 63. x 9and 5 65. x 3and 2 5 2 2 4 67.x 3and 69. x and 71. x and 4 3 5 5 5 1 73. no solution 75. x 2 Lesson 1.7A Solving linear inequalities and the types of notation Objective: To be able to use interval notation when solving linear inequalities, recognize inequalities with no solution or all real numbers as a solution. The set of all solutions is called the solution set of the inequality. Set-builder notation and a new notation called interval notation, are used to represent solution sets. (See Handout!) Interval Notation: See Handout Example 1: (-1,4] Graph and write the solution in set-builder notation. ( ] -1 0 Interval Notation (-1,4] = 4 Set Builder Notation {x -1 < x ≤ 4} The answer is read x such that -1 is less than x which is less than or equal to 4. You Try: (-∞, -4) ) -4 (-∞, -4) = { x x < -4} 0 Graphing intervals on a number line! Example 2: Graph each interval on a number line. (2,3] ( ] 2 3 Write answer using set builder notation. {x 2 < x < 3} You try: Graph and write in set builder notation. [1,6) [ ) 1 6 (-3,7] ( -3 ] 7 {x 1 < x < 6} {x -3 < x < 7} Example 3: Solve and graph this linear inequality. -2x - 4 > x + 5 Remember to switch the sign when you multiply or divide by a negative. x < -3 ) -3 0 You Try!! 3x+1 > 7x – 15 Use interval notation to express the solution set. Graph the solution. Answer: (-∞, 4) ) 4 Inequalities with Unusual Solution Sets Some inequalities have no solution. Example 4: x > x+1 There is no number that is greater than itself plus one. The solution set is an empty set ( this is a zero with a slash through it) Notes Continued: Like wise some inequalities are true for all real numbers such as: Example 5: x<x+1. Every real number is less than itself plus 1. The solution set is { x x is a real number} Interval notation, ( , ) or all real numbers. Notes Continued: When solving an inequality with no solution, the variable is eliminated and there will be a false solution such as 0 > 1. When solving an inequality that is all real numbers, the variable is eliminated and there will be a true solution such as 0 < 1. Example 6: Solving a linear inequality for a solution set. A. 2 (x + 4) > 2x + 3 B. x + 7 < x – 2 solution: 7 < -2 solution: 8 > 3 The inequality 8>3 is true for all values of x. The solution set is {x x is a real number} or , The inequality 7 < -2 is false for all values of x. The solution set is You try: 3(x + 1) > 3x + 2 3x + 3 > 3x + 2 3 > 2 The solution set is all real numbers. (-∞,∞) Summary: Describe the ways in which solving a linear inequality is different from solving a linear equation.