Unit Essential Questions How do you represent relationships between quantities that are not equal? Can inequalities that appear to be different be equivalent? How can you solve inequalities? Williams Math Lessons
TARGET INEQUALITIES AND THEIR GRAPHS MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.912.A-CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/ or inequalities, and interpret solutions as viable or non-viable options in a modeling context. RATING 4 3 2 1 LEARNING SCALE write, graph, and identify solutions of inequalities in real world situations or more challenging problems that I have never previously attempted write, graph, and identify solutions of inequalities write, graph, and identify solutions of inequalities with help understand the definition of an inequality WARM UP You show your student identification at a local restaurant in order to receive a 10% discount. You spend $18 for your meal. How much is the meal without the discount? KEY CONCEPTS AND VOCABULARY WRITING AND GRAPHING INEQUALITIES Symbols Words Graph x > 4 x is greater than 4 x 4 x is greater than or equal to 4 x < 4 x is less than 4 x 4 x is less than or equal to 4 EXAMPLES EXAMPLE 1: WRITING INEQUALITIES What inequality represents the verbal expression? a) The product of 12 and a number is less than 6. b) The sum of a number and 2 is no less than the product of 9 and the same number. -45-
EXAMPLE 2: IDENTIFYING SOLUTIONS TO AN INEQUALITY Is the number a solution of 3x 2 > 5? a) 2 b) 3 c) 0 EXAMPLE 3: GRAPHING AN INEQUALITY Graph the inequalities. a) x 3 b) x < 5 c) 4 x EXAMPLE 4: WRITING AN INEQUALITY FROM THE GRAPH What inequality represents the graph? a) b) EXAMPLE 5: WRITING INEQUALITIES IN REAL-WORLD SITUATIONS Write an inequality that describes the situation. a) b) RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 2 1-46-
SOLVING INEQUALITIES USING ADDITION OR SUBTRACTION MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.912.A-CED.A.1: Create equations or inequalities in one variable and use them to solve problems. MACC.912.A-CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/ or inequalities, and interpret solutions as viable or non-viable options in a modeling context. TARGET RATING 4 3 2 1 WARM UP Solve. LEARNING SCALE use addition and subtraction to solve inequalities in real-world situations or more challenging problems that I have never previously attempted use addition and subtraction to solve inequalities use addition and subtraction to solve inequalities with help understand that one can use properties of inequality to solve inequalities a) x 4 = 7 b) x 6 = 4 c) f + 5 = 0 d) y + 1 3 = 4 9 KEY CONCEPTS AND VOCABULARY ADDITION PROPERTY OF INEQUALITY If the same number is added to each side of a true inequality, the resulting inequality is also true. Let a, b, and c be real numbers. If a > b, then a + c > b + c If a < b, then a + c < b + c SUBTRACTION PROPERTY OF INEQUALITY If the same number is subtracted from each side of a true inequality, the resulting inequality is also true. Let a, b, and c be real numbers. If a > b, then a c > b c If a < b, then a c < b c EXAMPLES EXAMPLE 1: SOLVING INEQUALITIES USING THE ADDITION PROPERTY Solve. Graph each solution. a) x 12 > 8 b) 4 + x 5-47-
c) 22 > x 8 d) 14 x 19 EXAMPLE 2: SOLVING INEQUALITIES USING THE SUBTRACTION PROPERTY Solve and graph each solution. a) y + 3 < 7 b) a + 10 12 c) d + 4 7 > 1 14 d) w + 2.8 13.9 e) 9 5 + r f) 16 < h + 12 EXAMPLE 3: SOLVING INEQUALITIES IN REAL-WORLD SITUATIONS You have been saving for a new phone that costs no less than $200. You have saved $130. Write and solve an inequality to determine that amount you need to purchase the phone. RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 2 1-48-
TARGET SOLVING INEQUALITIES USING MULTIPLICATION OR DIVISION MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.912.A-CED.A.1: Create equations or inequalities in one variable and use them to solve problems. RATING 4 3 2 1 LEARNING SCALE use multiplication and division to solve inequalities in real-world situations or more challenging problems that I have never previously attempted use multiplication and division to solve inequalities use multiplication and division to solve inequalities with help understand that one can use properties of inequality to solve inequalities WARM UP Describe and correct the error in solving each inequality. 1) Solve. 2) Solve. 3+ x > 1 2 x + 5 3+ x + 3 > 1 3 x > 2 KEY CONCEPTS AND VOCABULARY 2 5 x + 5 5 x 3 MULTIPLICATION PROPERTY OF INEQUALITY If the same positive number is multiplied to each side of a true inequality, the resulting inequality is also true. Let a, b, and c be real numbers with c > 0. If a > b, then ac > bc If a < b, then ac < bc If the same negative number is multiplied to each side of a true inequality, the inequality sign is reversed to make the resulting inequality true. Let a, b, and c be real numbers with c < 0. If a > b, then ac < bc If a < b, then ac > bc DIVISION PROPERTY OF INEQUALITY If both sides of a true inequality are divided by the same positive number, the resulting inequality is also true. Let a, b, and c be real numbers with c > 0. If a > b, then a/c > b/c If a < b, then a/c < b/c If both sides of a true inequality are divided by the same negative number, the inequality sign is reversed to make the resulting inequality true. Let a, b, and c be real numbers with c < 0. If a > b, then a/c < b/c If a < b, then a/c > b/c EXAMPLES EXAMPLE 1: SOLVING INEQUALITIES USING THE MULTIPLICATION PROPERTY (POSITIVE NUMBER) Solve. a) x 4 3 b) x 7 > 2 c) 1 3 x < 2 3 d) 5 x 2-49-
EXAMPLE 2: SOLVING INEQUALITIES USING THE MULTIPLICATION PROPERTY (NEGATIVE NUMBER) Solve. a) x 5 > 4 b) x 2 6 c) x 4 < 3 8 d) 4 1 3 x EXAMPLE 3: SOLVING INEQUALITIES USING THE DIVISION PROPERTY (POSITIVE NUMBER) Solve. a) 2m 6 b) 4a < 12 c) 7p > 2 d) 9 18g EXAMPLE 4: SOLVING INEQUALITIES USING THE DIVISION PROPERTY (NEGATIVE NUMBER) Solve. a) 5t 55 b) 9x > 72 c) 63n < 7 d) 144 12k EXAMPLE 5: SOLVING INEQUALITIES IN REAL-WORLD SITUATIONS You work at a local car wash and make $8.25 per hour. How many hours do you need to work to make at least $165? RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 2 1-50-
TARGET SOLVING MULTI-STEP INEQUALITIES MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.912.A-CED.A.1: Create equations or inequalities in one variable and use them to solve problems. RATING 4 3 2 1 LEARNING SCALE solve multi-step inequalities in real-world situations or more challenging problems that I have never previously attempted solve multi-step inequalities solve multi-step inequalities with help understand that one solves a multi-step inequality using properties of inequalities WARM UP Solve each equation. 1) 5x + 7 2x = 11 2) 3(t + 7) 4 = 17 3) 0.25m + 0.5 + 0.75m = 1.5m KEY CONCEPTS AND VOCABULARY can be solved the same way you would solve a multi-step equation. Just remember that when you multiply or divide by a negative number, the inequality symbol changes. EXAMPLES EXAMPLE 1: SOLVING MULTI-STEP INEQUALITIES Solve each inequality. Graph the solutions. a) 3x - 8 > 1 b) 3v 5v + 18 EXAMPLE 2: SOLVING MULTI-STEP INEQUALITIES WITH NEGATIVE COEFFICIENTS Solve each inequality. Graph the solutions. a) 7 x 24 b) 62 < 3x + 2-51-
EXAMPLE 3: SOLVING MULTI-STEP INEQUALITIES USING THE DISTRIBUTIVE PROPERTY Solve each inequality. Graph the solutions. a) 2(y 3) + 7 < 21 b) 2(x + 11) > 3(x + 6) EXAMPLE 4: WRITING AND SOLVING MULTI-STEP INEQUALITIES Write an inequality and solve the problem. a) Twelve added to a number is less than twice the number minus eight. b) Three more than half a number is less than fifteen. EXAMPLE 5: SOLVING INEQUALITIES WITH SPECIAL SOLUTIONS Solve each inequality. a) 2(3x + 1) > 6x + 7 b) 5(2x 3) 7x 3x + 8 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 2 1-52-
TARGET SOLVING COMPOUND INEQUALITIES MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.912.A-CED.A.1: Create equations or inequalities in one variable and use them to solve problems. RATING 4 3 2 1 LEARNING SCALE solve compound inequalities in real-world situations or more challenging problems that I have never previously attempted solve compound inequalities containing the word and solve compound inequalities containing the word or solve compound inequalities containing the word and with help solve compound inequalities containing the word or with help understand the definition of a compound inequality WARM UP Determine if the inequality is always, sometimes, or never true. 1) 6x + 8 8 + 6x 2) 7x + 3 > 7x 2 3) 4(x 2) < 4x 10 KEY CONCEPTS AND VOCABULARY - two inequalities joined with the word and or the word or AND means that a solution makes BOTH inequalities true. OR means that a solution makes EITHER inequality true. EXAMPLES EXAMPLE 1: WRITING COMPOUND INEQUALITIES Write a compound inequality that represents the phrase. Graph the solutions. a) All real numbers less than four and greater than negative two. b) All real numbers less than or equal to 1 or greater than or equal to six. -53-
EXAMPLE 2: SOLVING COMPOUND INEQUALITIES INVOLVING AND Solve each compound inequality. Graph the solution. a) 4r > 12 and 2r < 10 b) z + 3 1 and z 2 < 1 c) 2 < x + 1 < 4 d) 3 < 5x 2 < 13 EXAMPLE 3: SOLVING COMPOUND INEQUALITIES INVOLVING OR Solve each compound inequality. Graph the solution. a) 3x < 6 or 7x > 35 b) p + 5 10 or 2p > 10 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 2 1-54-