Section 6.1: Solving Inequalities by Addition and Subtraction How do we solve the equation: x 12 = 65? How do we solve the equation: x 12 < 65? Graph the solution: Example 1: 12 y 9 Example 2: q + 23 < 14 Example 3: 12n 4 13n Example 4: 5 p + 7 > 6 p -1-
Example 5: Seven times a number is greater than six times that number minus two Example 6: A number decreased by 8 is at most 14-2-
Section 6.2: Solving Inequalities by Multiplication and Division 8 > 5 8 > 5 If each side of an inequality is multiplied by a of the inequality symbol is changed. number, the 15 > 5 15 > 5 If each side of an inequality is divided by a of the inequality symbol is changed. number, the Example 1: 3 g < 12 Example 2: 3 g < 12 Example 3: 1 x 5 4-3-
Example 4: 3 x 6 4 Example 5: One-fourth of a number is less than -7 Example 6: Four-fifths of a number is at most 20 Example 7: Negative five times a number is less than or equal to ten -4-
Section 6.3: Solving Multi-Step Inequalities Example 1: 13 11d 79 Example 2: 4x + 12 < x 3 Example 3: ( c + 3) 6c + 3( c) 8 2-5-
Example 4: ( x + 4) + 11x 8x 2( 2x 1) 7 + Example 5: 7n 1+ 2n 29 Example 6: 5b + 8 < 3b 3-6-
Example 7: Seven times a number is less than or equal to two less than six times that number. Example 8: What value for x will yield an area at least 58 square inches if the base is 8 inches long and the height is (x + 2) inches? Example 9: Ty has $80 to spend shopping. If he spends $42 on pants and $29 on a shirt, how much can he spend on a belt? -7-
Example 10: In the first 3 days of the week Isabella earned $90 in tips, but by the end of the week her total for tips exceeded $250. She earned at least how much the last 4 days of the week? Example 11: The car rental company charges $40 per day plus $0.10 a mile. How many miles can you drive and keep your bill at $100 or less for a two day rental period? Example 12: The volume of a rectangular fish aquarium cannot exceed 3456 cubic inches. If the width and height are both 12 inches, the length is at most how much? -8-
Section 6.4: Solving Compound Inequalities A compound inequality is inequality connected with the words or. True or False? 5 < 4 or 8 < 6 1 > 0 and 1 < 5 4 > 0 and 4 < 0 0 = 0 or 2 > 3 5 5 or 1 > 4 0 > 3 and 2 > 2 The word means that must be true. The word mean that must be true. is an intersection and is a union. Example 1: y 5 and y < 12 Example 2: 7 < z + 2 11-9-
TIP: When you have a compound equation of the form: a< x< b, you can break it into 2 equations: and Example 3: k + 2 > 12 and k + 2 18 Example 4: 8> 5 3q and 5 3q > 13 Example 5: m 4 or m > 6-10-
Example 6: 3n + 11 13 or 3n 12 Example 7: 4c < 2c 10 or 3c < 12 Example 8: t 1 or t 3-11-
Example 9: 6 2z 4 Example 10: 9 2a + 5 < 15 Example 11: ( x ) 2 3-12-
Example 12: ( x ) 6 3 2 + 4 3 Example 13: 3x 7 < 11 or 9x 4 > x + 4 Example 14: 4 3x 8 or 3x 1 8-13-
6.5: Solving Open Sentences Involving Absolute Value x = 5 means. x < 5 means. x > 5 means. x 6 = 2 b + 6 = 5 The points and are units from the point. Write an absolute value equation for: - Find the point from -4 and 2. - Determine the distance of each point from. - Write the equation:. -14-
Solve x 3 = 12 : Algebra 1 b + 6 > 5 b + 6 < 5 Example 1: x 3 12 When you solve an absolute value inequality of the type, you are finding the of 2 inequalities. When you solve an absolute value inequality of the type, you are finding the of 2 inequalities. Example 2: x + 2 > 5-15-
Example 3: 3y 3 > 9 Example 4: 2 x + 3 < 8 Graphing Absolute Value Equations in 2 variables y = x X -2 Y -1 0 1 2-16-
y = x 2 + 3 X 0 Y 1 2 3 4 We call the point of the absolute value function graph the. We can find the vertex by taking the expression inside the absolute value and setting it equal to. Absolute value function graphs have which means that they look the same on both sides of the. Example 1: y = x+ 3 1 X Y -17-
Example 2: y = x + 3 X Y Example 3: y = 2 x X Y -18-
Example 4: y = x X Y Example 5: y = 2 x+ 3 5 X Y -19-
Example 6: y = x 2 + 1 X Y -20-
6.6: Graphing Inequalities in Two Variables 4x+ 2y > 8 Replacement set: {( 3,3 ),( 0, 2 ),( 2, 4 ),( 1,0 )} Solution set: The is bounded (or separated) by the. When graphing a linear inequality, we use a for < and >. for or and a -21-
Example 1: Algebra 1 { } y 3 2x ( 0, 4 ),( 1,3 ),( 6, 8 ),( 4,5) Example 2: { } x+ y < 11 ( 5,7 ),( 13,10 ),( 4, 4 ),( 6, 2) -22-
Example 3: 2y+ x 6 Steps to Graph a Linear Inequality 1. 2. 3. -23-
Example 4: 8x 6y < 12 Example 5: 3x 1 y -24-