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Chapter 2 Equations, Inequalities, and Applications Section 2.1 – The Addition Property of Equality Homework problems: 1-3, 7-10, 25-27, 36-38, 46-48, 51-53 Additional Problems Solve: 1) x + 5 = 10 2) x – 5 = 10 3) 5y + 12 = 4y 4) 7y = 6y + 12 5) 3z + 5 = 4z – 6 6) 3a + 2 + 4a = 11 + 8a – 2 7) 5(2b + 3) – 3(b – 2) = 4(b – 2) + 2(b + 4) Page 1 of 20 Chapter 2 Equations, Inequalities, and Applications Section 2.2 – The Multiplication Property of Equality Homework problems: 1-3, 11-13, 17, 25-27, 43-45 Additional Problems Solve: 1) 7x – 3 = 4x + 6 2) 4(3x – 8) + 7 = 5(3 – 5x) – 2 3*) 4(3x – 8) = 6(2x + 1) 4) 1.5x – 3.4 = 5.9 – 1.6x 5) 2 3 𝑦 = 8 * This problem is tricky Page 2 of 20 Chapter 2 Equations, Inequalities, and Applications Section 2.3 – More on Solving Linear Equations Homework problems: 17-19, 31-35, 45-47, 61, 63, 65, 67 Additional Problems Solve: 1) 7x – 4 = 3x + 6 2) 12x – 6 = 6(2x – 1) 3) 4 15 2 2 3 𝑠 + 5 = 21 𝑠 + 7 4) .14(3x – 2) – 1 = .29(3 – 2x) + 2 Write the expression in terms of the given variable: 5) Two more than the number a. 6) The number whose product with b is 10. 7) The number whose sum with c is 10. Page 3 of 20 Chapter 2 Equations, Inequalities, and Applications Page 4 of 20 Section 2.4 – An Introduction to Applications of Linear Equations Homework problems: 15-16, 21-22, 27-28, 35-36 Additional Problems Solve: 1) Grandpa is currently 6 times as old as Susie. In six years he will be only 4 times as old. How old is Susie and grandpa now? 2) Betty normally gets 30 miles per gallon in her Mazda. If she traveled 315 miles between fillups, how much gas does she think her car needs? Chapter 2 Equations, Inequalities, and Applications Section 2.5 – Formulas and Additional Applications from Geometry Homework problems: 13-14, 19-20, 29-30, 63-64, 77-78 Additional Problems 𝑃𝐵 1) For the formula 𝐴 = 100: a) Solve for P b) Solve for B c) Given A = 150 and P = 20, find B 𝑎𝑏+𝑐 2) For the formula 𝑇 = 𝑑+10 a) Solve for a b) Solve for c c) solve for d d) Given a = 5, b = 6, c = 10, and d = 10 find T. Page 5 of 20 Chapter 2 Equations, Inequalities, and Applications Page 6 of 20 Section 2.6 – Ratio, Proportion, Percent Homework problems: 1, 15-17, 27-29, 39-41, 49-52 Additional Problems 1) John is 6 feet tall. During the day, someone measured his shadow and it was 2 feet long. At the same time, someone else measured the shadow of a building and it was 40 feet long. How tall was the building? 2) Jerry was paid 1% interest on the minimum balance of his savings every three months. (That is 4% APY). How much interest did he earn on a minimum balance of $50? 3) If the sales tax is 7%, what was the price before sales tax on an item that cost $42.80 with sales tax included? Chapter 2 Equations, Inequalities, and Applications Section 2.7 – Solving Linear Inequalities Homework problems: 1-6, 8-9, 12-13, 17-20, 49-52 Additional Problems 1) Graph 4 < 𝑥 ≤ 11 below 2) Graph 7 > 𝑥 below 3) Graph 2 ≤ 𝑥 < 7 below 4) Graph 1 ≤ 𝑥 below Solve the inequalities below. Write the answers in set interval notation. 5) 5𝑝 < 25 6) −4𝑥 + 3 ≥ 19 7) 7 < 3𝑥 − 5 < 22 8*) 10 ≥ 6 − 2𝑥 ≥ −10 * A little different than in the book Page 7 of 20 Chapter 2 Equations, Inequalities, and Applications Vocabulary 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) areas circles combine conditional (equation) contradiction converting cross (products) distributive (property) empty (set) equation equivalent (equations) extremes graph greater identity inequality infinitely infinity intervals inverse less like linear (equation) literal (equation) means measurements null (set) percent percentages perimeters proportions ratio rectangle simplify solution (set) squares terms trapezoid triangle unknown Page 8 of 20 Chapter 2 Equations, Inequalities, and Applications Vocabulary by Section Section 1 combine distributive (property) equation equivalent (equations) identity inverse like linear (equation) simplify solution (set) terms Section 2 decimals divide equality multiplication variable Section 3 conditional (equation) contradiction empty (set) null (set) infinitely Section 4 unknown Section 5 areas circles converting literal (equation) measurements perimeters rectangle squares trapezoid triangle Section 6 cross (products) extremes means percentages percent proportions ratio Section 7 graph greater inequality infinity intervals less Page 9 of 20 Chapter 2 Equations, Inequalities, and Applications Page 10 of 20 Section 2.1 – The Addition Property of Equality Additional Answers Solve: 1) x + 5 = 10 x + 5 – 5 = 10 – 5 x=5 Subtract 5 from each side Answer (combine like terms) 2) x – 5 = 10 x – 5 + 5 = 10 + 5 x = 15 Add 5 to each side Answer (combine like terms) 3) 5y + 12 = 4y 5y + 12 – 4y = 4y – 4y y + 12 = 0 y + 12 – 12 = 0 – 12 y = – 12 Subtract 4y from each side combine like terms Subtract 12 from both sides Answer (combine like terms) 4) 7y = 6y + 12 7y – 6y = 6y + 12 – 6y y = 12 Subtract 6y from each side Answer (combine like terms) 5) 3z + 5 = 4z – 6 3z + 5 – 4z = 4z – 6 – 4z –z + 5 = – 6 –z + 5 – 5 = – 6 – 5 –z = – 11 z = 11 Subtract 4z from both sides Combine like terms Subtract 5 from both sides combine like terms Answer: Take opposites of both sides 6) 3a + 2 + 4a = 11 + 8a – 2 7a + 2 = 9 + 8a 7a + 2 – 8a = 9 + 8a – 8a 2–a=9 2–a–2=9–2 –a=7 a = –7 combine like terms Subtract 8a from both sides combine like terms subtract 2 from both sides combine like terms Answer: Take opposites of both sides 7) 5(2b + 3) – 3(b – 2) = 4(b – 2) + 2(b + 4) 5(2b) + 5(3) – 3(b) – 3(–2) = 4(b) + 4(–2) + 2(b) + 2(4) Distribute multiplication 10b + 15 – 3b + 6 = 4b – 8 + 2b + 8 Do multiplication 7b + 21 = 6b Combine like terms 7b + 21 – 6b = 6b – 6b Subtract 6b from both sides b + 21 = 0 combine like terms b + 21 – 21 = 0 – 21 subtract 21 from both sides b = –21 Answer: combine like terms Chapter 2 Equations, Inequalities, and Applications Page 11 of 20 Section 2.2 – The Multiplication Property of Equality Additional Answers Solve: 1) 7x – 3 = 4x + 6 7x – 3 – 4x = 4x + 6 – 4x 3x – 3 = 6 3x – 3 + 3 = 6 + 3 3x = 9 x=9 2) 4(3x – 8) + 7 = 5(3 – 5x) – 2 12x – 32 + 7 = 15 – 25x – 2 12x – 25 = 13 – 25x 12x – 25 + 25x = 13 – 25x + 25x 37x – 25 = 13 37x – 25 + 25 = 13 + 25 37x = 38 38 𝒙 = 37 37 Subtract 4x from both sides Combine like terms Add 3 to both sides Combine like terms Answer: Divide both side by 3 Distribute multiplication Combine like terms Add 25x to both sides Combine like terms Add 25 to both sides Combine like terms Answer: Divide both sides of equation by 3*) 4(3x – 8) = 6(2x + 1) 12x – 32 = 12x + 6 12x – 32 – 12x = 12x + 6 – 12x –32 = 6 No solution Distribute multiplication Subtract 12x from both sides Combine like terms Answer: Above equation is never true 4) 1.5x – 3.4 = 5.9 – 1.6x 1.5x – 3.4 + 1.6x = 5.9 – 1.6x + 1.6x 3.1x – 3.4= 5.9 3.1x – 3.4 + 3.4 = 5.9 + 3.4 3.1x = 9.3 x=3 Add 1.6x to both sides Combine like terms Add 3.4 to both sides of the equation Combine like terms Answer: Divide both side by 3.1 5) 2 𝑦 = 8 3 3 2 3 2 3 2 ∙ 𝑦 = y = 12 ∙8 Multiply both sides by 3/2 Answer: Do multiplication Chapter 2 Equations, Inequalities, and Applications Page 12 of 20 Section 2.3 – More on Solving Linear Equations Additional Answers Solve: 1) 7x – 4 = 3x + 6 7x – 4 – 3x = 3x + 6 – 3x 4x – 4 = 6 4x – 4 + 4 = 6 + 4 4x = 10 𝟏𝟎 𝟓 x= 𝟒 =𝟐 Subtract 3x from both sides Combine like terms Add 4 to both sides Combine like terms Answer: divide both sides by 4 (reduce) 2) 12x – 6 = 6(2x – 1) 12x – 6 = 12x – 6 Any number x Distribute multiplication Answer: (Above equation is an identity) 3) 4 15 2 2 3 𝑠 + 5 = 21 𝑠 + 7 4 2 2 3 105(15 𝑠 + 5) = 105(21 𝑠 + 7) 28𝑠 + 42 = 10𝑠 + 45 28s + 42 – 10s = 10s + 45 – 10s 18s + 42 = 45 18s + 42 – 42 = 45 – 42 18s = 3 𝟑 𝟏 s = 𝟏𝟖 = 𝟔 4) .14(3x – 2) – 1 = .29(3 – 2x) + 2 100[.14(3x – 2) – 1] = 100[.29(3 – 2x) + 2] 14(3x – 2) – 100 = 29(3 – 2x) + 200 52x – 28 – 100 = 87 – 58x + 200 52x – 128 = 287 – 58x 52x – 128 + 58x = 287 – 58x + 58x 110x – 128 = 287 110x – 128 + 128 = 287 + 128 110x = 415 𝟒𝟏𝟓 𝟖𝟑 x = 𝟏𝟏𝟎 = 𝟐𝟐 Multiply both side by 105 (LCD) Distribute multiplication Subtract 10s from both sides Combine like terms Subtract 42 from both sides Combine like terms Answer: Divide both sides by 18 (reduce) Multiply both sides by 100 (LCD) Distribute multiplication Distribute multiplication Combine like terms Add 58x to both sides Combine like terms Add 128 to both sides Combine like terms Answer: Divide both sides by 110 (reduce) Write the expression in terms of the given variable: 5) Two more than the number a. a+2 6) The number whose product with b is 10. 7) The number whose sum with c is 10. 𝟏𝟎 𝒃 10 – c Chapter 2 Equations, Inequalities, and Applications Page 13 of 20 Section 2.4 – An Introduction to Applications of Linear Equations Additional Answers Solve: 1) Grandpa is currently 6 times as old as Susie. In six years he will be only 4 times as old. How old is Susie and grandpa now? S = Susie’s age now G = 6S = grandpa’s age now S+6 = Susie’s age in 6 years G+6 = 4(S+6) = Grandpa’s age in 6 years 6S + 6 = 4(S+6) Replace Grandpa’s age (now) 6S + 6 = 4S + 24 Distribute multiplication 6S + 6 – 4S = 4S + 24 – 4S Subtract 4S from each side 2S + 6 = 24 combine like terms 2S + 6 – 6 = 24 – 6 Subtract 6 from both sides 2S = 18 Combine like terms S=9 Answer: Divide by 2 to get Susie’s age G = 6S = 54 Answer: Calculate Grandpa’ 2) Betty normally gets 30 miles per gallon in her Mazda. If she traveled 315 miles between fillups, how much gas does she think her car needs? g = gallons used M = miles driven = 30g 315 = 30g put in miles driven 10.5 = g Answer: divide equation by 30 Chapter 2 Equations, Inequalities, and Applications Page 14 of 20 Section 2.5 – Formulas and Additional Applications from Geometry Additional Answers 𝑃𝐵 1) For the formula 𝐴 = 100: a) Solve for P 𝑃= b) Solve for B 𝐵= c) Given A = 150 and P = 20, find B 𝐵= 100𝐴 𝐵 100𝐴 𝑃 100𝐴 𝑃 = 100(150) 20 = 15,000 20 = 𝟕𝟓𝟎 𝑎𝑏+𝑐 2) For the formula 𝑇 = 𝑑+10 a) Solve for a b) Solve for c c) Solve for d 𝑇(𝑑+10)−𝑐 𝑎= 𝑏 𝑐 = 𝑇(𝑑 + 10) − 𝑎𝑏 𝑎𝑏+𝑐 𝑑 = 𝑇 − 10 𝑎𝑏+𝑐 d) Given a = 5, b = 6, c = 10, and d = 10 find T. 𝑇 = 𝑑+10 = (5)(6)+(10) (10)+10 = 30+10 20 40 = 20 = 𝟐 Chapter 2 Equations, Inequalities, and Applications Page 15 of 20 Section 2.6 – Ratio, Proportion, Percent Additional Answers 1) John is 6 feet tall. During the day, someone measured his shadow and it was 2 feet long. At the same time, someone else measured the shadow of a building and it was 40 feet long. How tall was the building? 6 𝑥 = 40 2 Set up proportion 6(40) = 2𝑥 240 = 2𝑥 𝟏𝟐𝟎 𝐟𝐞𝐞𝐭 = 𝒙 Cross multiply Do multiplication Answer: (divide by 2) 2) Jerry was paid 1% interest on the minimum balance of his savings ever three months. (That is 4% APY). How much interest did he earn on a minimum balance of $50? (1)(50) 𝑃𝐵 𝐴= = = $. 50 100 100 3) If the sales tax is 7%, what was the price before sales tax on an item that cost $42.80 with sales tax included? $42.80 = P + .07P = 1.07P Formula for price after tax $40.00 = P Answer: divide both side by 1.07 Chapter 2 Equations, Inequalities, and Applications Page 16 of 20 Section 2.7 – Solving Linear Inequalities Additional Answers 1) Graph 4 < 𝑥 ≤ 11 below 2) Graph 7 > 𝑥 below 3) Graph 2 ≤ 𝑥 < 7 below 4) Graph 1 ≤ 𝑥 below Solve the inequalities below. Write the answers in set interval notation. 5) 5𝑝 < 25 p<5 Divide by 5 (-∞, 5) Answer (set interval notation) 6) −4𝑥 + 3 ≥ 19 −4𝑥 ≥ 16 𝑥≤4 (-∞, 4] Subtract 3 from both sides Divide by –4, reversing inequality Answer (set interval notation) 7) 7 < 3𝑥 − 5 < 22 12 < 3𝑥 < 27 4<𝑥<9 (4, 9) Add 5 to all three sides Divide all three sides by 3 Answer (set interval notation) 8*) 10 ≥ 6 − 2𝑥 ≥ −10 4 ≥ −2𝑥 ≥ −16 −2 ≤ 𝑥 ≤ 8 [-2, 8] Subtract 6 to all three sides Divide sides by -2, reversing inequality Answer (set interval notation) Chapter 2 Equations, Inequalities, and Applications Page 17 of 20 Glossary Additive Identity: See Identity. Additive Inverse: See Inverse. Adjacent angles: See Plane Geometry. Angle: See Plane Geometry. Area: See Plane Geometry. Circle: See Plane Geometry. Circumference: See Perimeter under Plane Geometry. Combine like terms: Terms: are things that are added or subtracted to each other. Like terms: are terms whose variable factors can be rearranged (put in alphabetical order for example) such that all the factors are the same raised to the same power. Examples: 2xy and 5yx are like variables. Both consist of the same variables; the numeric part can normally be ignored. 2xy and 2x2y are not like variables because the powers of x are different. 3xx and 5x2 are like variables since x2 is just shortcut notation for xx. Combining like terms: Like terms can be combined by just adding / subtracting their numerical coefficients. Example: For 2xy + 3x – 4y + 5x – 1xy; we first rearrange the terms using the commutative and associative laws of addition to put like terms together to get (2xy – 1xy) + (3x + 5x) – 4y. To make it easier to see what is going on, we factor out the ‘like’ part next to get (2 – 1)xy + (3 + 5)x – 4y. Finally we combine the numerical coefficients to get 1xy + 8x – 4y or xy + 8x – 4y. Complimentary Angles: See Plane Geometry. Conditional Equation: See Equation. Contradiction: (Proof by contradiction) is when assuming a list of things is true (or false) and end up implying that at least one of the things in the list is false (or true). Converting: is to change one form (of say a number) to another form with an equivalent value. 𝑎 𝑐 Cross Products: The equation 𝑏 = 𝑑 is equivalent to the equation 𝒂𝒅 = 𝒃𝒄. Notice that you multiplied the numerator on one side of the equation by the denominator on the other side and set it equal to the product the numerator on the other side of the equation times the denominator on the first side used. The two products not only cross sides of the equation (one factor from each side) they also cross the fraction bar line (one factor from each side). Distributive Property of Multiplication: Might be stated as multiply everything inside parenthesis by whatever is outside the parenthesis. Examples: 2a(3b + 5c) = 2a(3b) + 2a(5c) = 6ab + 10ac; (5x – 4y)(-3z) = (5x) (–3z) – (4y)(-3z) = - 15xz + 12xz. The figure below, from geometry, shows that the area of the rectangle (a + b)(c) is the sum of the areas of two smaller rectangles ac + bc. Element: See Set. Empty Set: See Set. Chapter 2 Equations, Inequalities, and Applications Page 18 of 20 Equation: is a mathematical statement that two expressions have the same value. Conditional Equation: is a statement that is not true for all values of all the variables (otherwise the equation is an identity). Equivalent Equations: are two (or more) equations having the same solution set. Solution Set: Is the set of numbers for which an equation evaluates as a true statement. Equation Identity: See Identity. Equivalent Equations: See Equation. Geometry: See Plane Geometry. Identity: Additive Identity: Zero (0). For any number x, x + 0 = x. Multiplicative Identity: One (1). For any number x, x1 = x. Equation Identity: An equation that is true for all numerical values if the variables. Example: a + b = b + a. Inverse: Additive Inverse: What you have to add to a number to get zero (the additive inverse). Example: The additive inverse of x is –x. Multiplicative Inverse: What you have to multiply a number by to get one (the 1 multiplicative inverse). Example: The multiplicative inverse of x is 𝑥 = 1⁄𝑥 . Note that zero (0) has no multiplicative inverse. Inverse Operation: A binary operation, ⊕, has the inverse operation ⊖ if for any pair of numbers a and b the fact that a ⊕ b = c implies that c ⊖ b = a whenever there is just one value a such that a ⊕ b = c (for the given b and c). Addition and Subtraction are the inverse operation of each other. Division is the inverse operation of Multiplication. Inverse Function: For any function f(x), g(x) is the inverse function of f(x) over the domain D if x = g(f(x)) for all x in D. It is possible for f(x) to have different inverse functions over different domains. Note: This is not really part of this chapter. Like terms: See Combine like terms. Linear Equation: is an equation with no more than one variable factor in each term and no fractions except for numerical coefficients. When the equation contains just two variables, the plot of the solution on a Cartesian graph is a straight line. Multiplicative Identity: See Identity Multiplicative Inverse: See Inverse Null Set: See Set. Perimeter: See Plane Geometry Plane Geometry: Mathematical relations about figures on a flat surface. Adjacent angles: See Vertical angles below. Note that adjacent angles are always supplementary angles. Angle: is a measure between two straight (half) lines that start from the same point. Once around the circle the angle changes by 360°. Half way around is 180° and the two half lines become a straight line. Area: Is the measure of the inside of a figure. Basically it counts how many “unit squares” fit (can be placed) inside the figure. The unit squares can be cut up to fit. Fractional unit squares are allowed. Chapter 2 Equations, Inequalities, and Applications Page 19 of 20 Circle: is a figure whose boundary (edge) is a fixed radius (distance) from the circle’s center. Circle Complimentary Angles: is a pair of angles whose sum is 90° (a right angle or half a straight line) Perimeter: is a measure of how long the boundary (edge) of a figure is. For a circle, the perimeter is also called the circumference. Rectangle: is a four sided figure whose opposite sides are of equal length and parallel to each other and all four interior angles are equal (to 90°). Rectangle Square: is a four sided figure whose four sides are of equal length and all four interior angles are equal (to 90°). Square Supplementary angles: Are a pair of angles whose sum is 180° - a straight line. Triangle: is a figure with three sides (line segments). The sum of the interior angles of a triangle is 180°. Note: A convex polygon with n sides can always be split into (n – 2) triangles. The sum of the interior angles of a polygon with n sides is (n – 2) 180°. Triangle Trapezoid: is a four sided figure with a pair of sides parallel to each other. The distance between the parallel lines is referred to as the height of the trapezoid. Trapezoid Vertical Angles: Crossing straight lines create vertical (opposite angles) and adjacent angles. In the figure to the right angles “1 and 3” and angles “2 and 4” are vertical angles. Angles “1 and 2”, “2 and 3”, “3 and “4”, and “4 and 1” are adjacent angles. Note” Vertical angles always have equal measure. Vertical Angles Rectangle: See Plane Geometry. Set: is a collection of things. Element: is the generic name for something in a set. Empty Set: Is a set that contains no elements (is “empty”). Abbreviated as or {}. Null Set: is a synonym for the empty set. Chapter 2 Equations, Inequalities, and Applications Page 20 of 20 Set-builder notation: Shows the form of the elements in the set and describes all the members of the set. Example: {(x, y) | x is an even integer and y is an odd integer}. This is read as “The set of element pairs x, y where x is an even integer and y is an odd integer”. Note the “|” is read as “where”. Set Interval notation: Is a notation to indicate a connected set of real numbers. They are: (a, b) = {x | a < x < b} (a, b] = {x | a < x ≤ b} [a, b) = {x | a ≤ x < b} [a, b] = {x | a ≤ x ≤ b} Note: (a, a) = (a, a] = [a, a) = . [a, a] = {a} For a > b, (a, b) = [a, b) = (a, b] = [a, b] = . Set List Notation: is just a comma separated list of things inside braces. Example: {1, 2, 3, 8}. Set-builder notation: See Set. Set Interval notation: See Set. Set List Notation: See Set. Solution Set: See Equation. Square: See Plane Geometry. Supplementary angles: See Plane Geometry. Terms: See Combine like terms. Triangle: See Plane Geometry. Trapezoid: See Plane Geometry. Vertical Angles: See Plane Geometry.