Chapter 3: Inequalities 3-1: Graphing and Writing Inequalities Objectives: Identify solutions of inequalities in one variable. Write and graph inequalities in one variable. Inequality: The quantities are compared by using the following signs: Solution of an inequality: Identifying Solutions of Inequalities Example 1: Describe the solutions of x 6 4 in words. x - 3 0 9.9 10 10.1 12 x - 6 x 6 4 Solution? An inequality like 3 + x < 9 has too many solutions to list. You can use a graph on a number line to show all the solutions. The solutions are shaded and an arrow shows that the solutions continue past those shown on the graph. To show that an endpoint is a solution, draw a circle at the number. To show an endpoint is not a solution, draw an circle. Chapter 3 Page 1
Graphing Inequalities Graph each inequality. 2A: m ¾ 2a: c > 2.5 2b: 2 2 4 w 2B: t < 5(-1 + 3) 2c: m -3 Writing an Inequality from a Graph Write the inequality shown by each graph 3A: 3a: 3B: means Symbol: means Symbol: Application Example 4: Ray s dad told him not to turn on the air conditioner unless the temperature is at least 85 F. Define a variable and write an inequality for the temperatures at which Ray can turn on the air conditioner. Graph the solutions. Let t represent the temperatures at which Ray can turn on the air conditioner. Homework: Sec 3-1 (pg 171) 6, 7, 13, 14, 22, 23, 26-33, 50-53, 70, 74, 79, 80 (Due in 2 class periods on paper: 6, 7, 22, 23, 32, 33) Chapter 3 Page 2
3-2: Solving Inequalities by Adding or Subtracting Objective: Solve one-step inequalities by using addition. Solve one-step inequalities by using subtraction. Solving one-step inequalities is much like. To solve an inequality, you need to isolate the variable using the and. Use an inverse operation to the operation in an inequality. If the inequality contains addition, use to undo the addition. Using Addition and Subtraction to Solve Inequalities 1A: x + 12 < 20 1a: s + 1 10 1B: d 5 > 7 1b: 2.5 > 3 + t Homework: Sec 3-2: (Pg 177) 1, 3, 7-10, 13-23, 26-29, 38, 53, 56 (Due in 2 class periods on paper: 1, 3, 7-10, 13-23 GRAPHS. Still type inequality answers in online.) Chapter 3 Page 3
3-3: Solving Inequalities by Multiplying or Dividing Objective: Solve one-step inequalities by using multiplication. Solve one-step inequalities by using division. Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, the operation by dividing or multiplying both sides of the inequality by the same number. Multiplying or Dividing by a Positive Number 1A: 7x > 42 1a: 4k > 24 1B: 2.4 m 3 1b: 50 5q 3 1C: r < 12 4 3 1c: g > 27 4 If you multiply or divide both sides of an inequality by a number, the resulting inequality is not a true statement. You need to to make the statement true. Chapter 3 Page 4
Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < 24. Multiplying or Dividing by a Negative Number 2A: 12x > 84 2a: 10 x 2B: 8 x 3 2b: 4.25 > 0.25h Application Example 3: Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy? Let p represent the number of tubes of paint that Jill can buy. Homework: Sec 3-3: (Pg 183) 1, 3, 10, 18, 20, 22, 26, 29, 30, 32, 39, 41, 42, 51-54, 81, 84 (42 and 81 DO NOT need graphs; all other problems DO need graphs. Graphs are due in 2 class periods on paper. Still type inequality answers in online.) Chapter 3 Page 5
3-4: Solving Two-Step and Multi-Step Inequalities Objective: Solve inequalities that contain more than one operation. Inequalities that contain more than one operation require. Use to the operations in the inequality one at a time. Solving Multi-Step Inequalities 1A: 45 + 2b > 61 1a: 12 3x + 6 1B: 8 3y 29 x + 5 1b: > 3 2 To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides by using the order of operations, combining like terms, or using the Distributive Property. Simplifying Before Solving Inequalities Solve the inequality and graph the solutions 2A: 2 ( 10) > 4t 2a: 2m + 5 > 5 2 2B: 4(2 x) 8 2b: 3 + 2(x + 4) > 3 Homework: 3-4 (Pg 191) 1, 5, 9, 16-22, 28-33, 38, 40, 43, 49, 51, 52, 78, 79 (78 and 79 DO NOT need graphs; all other problems DO need graphs. Graphs are due in 2 class periods on paper. Still type inequality answers in online.) Chapter 3 Page 6
3-5: Solving Inequalities with Variables on Both Sides Objectives: Solve inequalities that contain variable terms on both sides. Some inequalities have variable terms on both sides of the inequality symbol. You can like you solved equations with variables on both sides. Use the properties of inequality to all the variable terms on one side and all the constant terms on the other side. 3 STEPS TO SOLVING: 1._ 2._ 3._ Solving Inequalities with Variables on Both Sides 1A: y 4y + 18 1a: 4m 3 < 2m + 6 1B: 4m 3 < 2m + 6 1b: 1b: 5t + 1 < 2t 6 Example 2: Skip You may need to simplify one or both sides of an inequality before solving it. Look for and places to use the. Chapter 3 Page 7
Simplify Each Side Before Solving 3A: 2(k 3) > 6 + 3k 3 3a: 5(2 r) 3(r 2) 3B: 0.9y 0.4y 0.5 3b: 0.5x 0.3 + 1.9x < 0.3x + 6 There are special cases of inequalities called and. Identities and Contradictions Solve the inequality. 4A: 2x 7 5 + 2x 4a: 4(y 1) 4y + 2 4B: 2(3y 2) 4 3(2y + 7) 4b: x 2 < x + 1 Homework: 3-5 (Pg 197) 3, 7, 10, 14, 15, 21, 24, 25, 28, 30, 31, 34, 36, 37, 41, 43, 44, 52, 53 (14, 15, 36, 37, 52, 53 DO NOT need graphs; all other problems DO need graphs. Graphs are due in 2 class periods on paper. Still type inequality answers in online.) Chapter 3 Page 8