Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 2 Equations, Inequalities and Problem Solving
22 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. The sum of the angles of a triangle is 180°. If one angle of a triangle measures x° and a second angle measures (2x+7)°, express the measure of the third angle in terms of x. Simplify the expression. 2. A quadrilateral is a four-sided figure whose angle sum is 360°. If one angle measures x°, another 3x°, and another 5x°, express the measure of the fourth angle in terms of x. Simplify the expression. Hint: DRAW THE PICTURE!!
33 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. The sum of the angles of a triangle is 180°. If one angle of a triangle measures x° and a second angle measures (2x+7)°, express the measure of the third angle in terms of x. Simplify the expression. x° (2x+7)° ?°
44 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. The sum of the angles of a triangle is 180°. If one angle of a triangle measures x° and a second angle measures (2x+7)°, express the measure of the third angle in terms of x. Simplify the expression. x° (2x+7)° ?° ?° = 180° - x° - (2x+7)° = (-3x+173)°
55 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 2. A quadrilateral is a four-sided figure whose angle sum is 360°. If one angle measures x°, another 3x°, and another 5x°, express the measure of the fourth angle in terms of x. Simplify the expression. x° ?° 5x° 3x°
66 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 2. A quadrilateral is a four-sided figure whose angle sum is 360°. If one angle measures x°, another 3x°, and another 5x°, express the measure of the fourth angle in terms of x. Simplify the expression. x° ?° 5x° 3x° ?° = 360° - x° - 3x° - 5x° = (-9x+360)°
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 2.3 The Multiplication Property of Equality
88 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Objectives: Use the multiplication property of equality to solve linear equations Write work phrases as algebraic expressions
99 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Multiplication Property of Equality Let a, b, and c represent numbers and let c ≠ 0. Then, a = ba = b and a · c = b · cand are equivalentare equivalent equations.equations.
10 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1
11 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1
12 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1
13 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1
14 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 A number divided by itself is one!
15 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
16 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
17 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
18 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
19 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
20 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
21 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one!
22 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one! True Statement!
23 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve –4x = 16 for x. Check: Example 1 1 A number divided by itself is one! ✔ True Statement!
24 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2
25 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2
26 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2
27 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2 The product of reciprocals is one!
28 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2 The product of reciprocals is one!
29 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2 The product of reciprocals is one! Simplify
30 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example 2 The product of reciprocals is one! Simplify
31 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: –1.2x = –36 Example 3
32 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: –1.2x = –36 Example 3
33 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: –1.2x = –36 Example 3
34 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: –1.2x = –36 Example 3 A number divided by itself is one!
35 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: –1.2x = –36 Example 3 A number divided by itself is one!
36 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: –1.2x = –36 Example 3 A number divided by itself is one!
37 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve:
38 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve:
39 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve:
40 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve:
41 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve:
42 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve: The product of reciprocals is one!
43 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 4 Simplify both sides. Multiply both sides by 7. Solve: The product of reciprocals is one!
44 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 4x – 8x = 16 Example 5
45 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 4x – 8x = 16 Example 5
46 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 4x – 8x = 16 Example 5
47 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 4x – 8x = 16 Example 5
48 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 4x – 8x = 16 Example 5 A number divided by itself is one!
49 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 4x – 8x = 16 Example 5 A number divided by itself is one!
50 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26
51 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26
52 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides!
53 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides!
54 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides! Simplify
55 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides! Simplify
56 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides! Simplify Divide on both sides!
57 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides! Divide on both sides! Simplify
58 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides! Divide on both sides! Simplify
59 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 6 3z – 1 = 26 3z = 27 z = 9 3z – = Solve: 3z – 1 = 26 Add to both sides! Divide on both sides! Simplify
60 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7
61 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify
62 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify
63 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides!
64 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides!
65 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides! Simplify
66 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides! Simplify
67 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides! Simplify Divide on both sides!
68 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides! Simplify Divide on both sides!
69 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides! Simplify Divide on both sides! Simplify
70 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12x x – 6 = 10 20x + 24 = 10 20x = – 14 20x ( – 24) = 10 + ( – 24) Solve: Example 7 Simplify Add to both sides! Simplify Divide on both sides! Simplify
71 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8
72 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute
73 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute
74 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify
75 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify
76 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides!
77 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides!
78 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides! Simplify
79 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides! Simplify
80 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides! Simplify Divide on both sides!
81 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides! Simplify Divide on both sides!
82 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides! Simplify Divide on both sides! Simplify
83 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 5(2x + 3) = – (2x) + 5(3) = – x + 15 – 15 = 6 – 15 10x + 15 = 6 10x = –9 Example 8 Distribute Simplify Add to both sides! Simplify Divide on both sides! Simplify
84 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Recall: Integers are the whole steps from -∞ to ∞, and consecutive means one right after another. Consecutive Integers Consecutive integers If n is the first integer, then: n+1 is the second, n+2 is the third, n+3 is the fourth, etc…
85 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Recall: Integers are the whole steps from -∞ to ∞, and consecutive means one right after another. Consecutive Integers Consecutive even integers If n is the first integer, then: n+2 is the second, n+4 is the third, n+6 is the fourth, etc…
86 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Recall: Integers are the whole steps from -∞ to ∞, and consecutive means one right after another. Consecutive Integers Consecutive odd integers If n is the first integer, then: n+2 is the second, n+4 is the third, n+6 is the fourth, etc…
87 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9
88 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9
89 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second
90 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second express the sum
91 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second express the sum 1 st + 2 nd is the sum
92 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second express the sum 1 st + 2 nd is the sum x + (x + 1)
93 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second express the sum 1 st + 2 nd is the sum x + (x + 1) Simplify
94 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second express the sum 1 st + 2 nd is the sum x + (x + 1) x + x + 1 Simplify
95 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. If x is the first of two consecutive integers, express the sum of the first and the second integer in terms of x. Simplify if possible. Example 9 x is the first consecutive integer no even or odd, so x + 1 is the second express the sum 1 st + 2 nd is the sum x + (x + 1) x + x + 1 Simplify 2x + 1
96 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: 1. What is the Multiplication Property of Equality? 2. What should you do to both sides of an equation?
97 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass 5(3x – 1) + 2 = 12x + 6 Step 1_________________ Step 2_________________ Step 3_________________ Step 4_________________ Step 5_________________ Solve the equation for x. Describe your process for each step.
98 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass Solve the equation for x. Describe your process for each step. 5(3x – 1) + 2 = 12x x – = 12x + 6 _________________ 15x – 3 = 12x + 6 _________________ 3x – 3 = 6 _________________ 3x = 9 _________________ 3x/3 = 9/3 _________________ x=3
99 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass Solve the equation for x. Describe your process for each step. 5(3x – 1) + 2 = 12x x – = 12x + 6 _________________ 15x – 3 = 12x + 6 _________________ 3x – 3 = 6 _________________ 3x = 9 _________________ 3x/3 = 9/3 _________________ x=3 Distribute
100 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass Solve the equation for x. Describe your process for each step. 5(3x – 1) + 2 = 12x x – = 12x + 6 _________________ 15x – 3 = 12x + 6 _________________ 3x – 3 = 6 _________________ 3x = 9 _________________ 3x/3 = 9/3 _________________ x=3 Distribute Simplify or Combine like terms
101 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass Solve the equation for x. Describe your process for each step. 5(3x – 1) + 2 = 12x x – = 12x + 6 _________________ 15x – 3 = 12x + 6 _________________ 3x – 3 = 6 _________________ 3x = 9 _________________ 3x/3 = 9/3 _________________ x=3 Distribute Simplify or Combine like terms Add same on both sides
102 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass Solve the equation for x. Describe your process for each step. 5(3x – 1) + 2 = 12x x – = 12x + 6 _________________ 15x – 3 = 12x + 6 _________________ 3x – 3 = 6 _________________ 3x = 9 _________________ 3x/3 = 9/3 _________________ x=3 Distribute Simplify or Combine like terms Add same on both sides
103 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Exit Pass Solve the equation for x. Describe your process for each step. 5(3x – 1) + 2 = 12x x – = 12x + 6 _________________ 15x – 3 = 12x + 6 _________________ 3x – 3 = 6 _________________ 3x = 9 _________________ 3x/3 = 9/3 _________________ x=3 Distribute Simplify or Combine like terms Add same on both sides Divide on both sides Add same on both sides