Download Chapter 2 Page of 20 Equations, Inequalities, and Applications

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, x1 = 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.