2.1 Solving Equations Using Properties of Equality Math ... · 2.1 Solving Equations Using Properties of Equality Math 085 Chapter 2 ... 2.5 Problem Solving ... 2.1 Solving Equations

2.1 Solving Equations Using Properties of Equality Math 085 Chapter 2

Chapter 2

2.1 Solving Equations Using Properties of Equality

2.2 More about Solving Equations

2.3 Application of Percent

2.4 Formulas

2.5 Problem Solving

2.6 More about Problem Solving

2.7 Solving Inequalitites

8.4 Solving Compound Inequalities

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Solving equations: A concrete example.

Suppose we have a barn that needs concrete on one side for animals. The place where

concrete is to be poured is 4 inches deep, 121

2feet wide and 122

1

2feet long. The amount of

concrete needed is on various pages - one page for each depth needed (like 1 in, 2 in, etc.).

To find what is needed we use the 4 inch page and then read the table. We need to find 121

2

feet wide and 1221

2feet long.

What if the length are only natural numbers?

What if the lengths are off the chart. How could you figure out how much concrete to order?

Real Number Properties

Given a, b, c ∈ R

Addition property of Equality a = b⇒ a + c = b + c

Subtraction property of Equality a = b⇒ a− c = b− c

Multiplication property of Equality a = b⇒ a • c = b • c

Division property of Equality a = b⇒ a

c=

b

cprovided c 6= 0

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Use the Real Number Properties of equality to solve.

Example 1: 2x− 7 = 21

2x = 28x = 14

Use the Real Number Properties of equality to solve.

x + 10 = 331

2x− 6 = 4

2x = 151

2x = 7

3

4x +

1

2x = 7 2.2− 3.3x = 4.4x + 5.5

8x + 7 = 7

Two numbers have a sum of 13. If one number is y, express the other in terms of y.

If x is the first of four consecutive even integers, express the sum of the first even integerand the third even integer as an algebraic expression containing the variable x.

A quadrilateral is a four sided figure like the one shown below whose angle sum is 360◦. Ifone angle measures x◦, a second angle measures 3x◦, and a third angle measures 5x◦, expressthe measure of the forth angle in terms of x. Simplify the expression.

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3x◦

?

5x◦

x◦

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2.2 More about Solving Equations Math 085 Chapter 2

An equation can be either TRUE or FALSE. If it contains a variable, then it may betrue for some values while false for others.

Examples:2 + 3 = 5 is true while 3 + 4 = 55 is false.

2 + x = 5 is true for x = 3, but false for all other values of x.

A solution is any number that when substituted in for a variable makes the equation aTRUE statement

A solution set is the set of all solutions (to a given problem).

Solving linear equations

Consider SOLVING an equation by using the following table:x −3 −2 −1 0 1 2 3 43x− 7 −16 −13 −10 −7 −4 −1 2 5

So if we then wanted the linear equation 3x−7 = −4 we could look up that to find x = 1

The only problem with this is that it’s extremely inefficient and tedious as we would needtables for ever equation that we might want to solve.

3x− 7+7 = −4+7

3x = 3

x =3

3x = 1

1

4x +

1

2=

5

6

Multiple by the LCD!!!(

12

1

) (

1

4x

)

+(

12

1

) (

1

2

)

=(

12

1

) (

5

6

)

3(1x) + 6(1) = 2(5)3x + 6 = 103x = 4

So x =4

3

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1

2x +

1

3=

1

6

x

8− 2 = 3

1

3x =

1

2

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What happens if we have a variable on both sides of the equal sign?

Move the variable to one side - that is isolate the variable.

5x +7 = 3x −4

5x +7 = 3x −4−3x −3x

2x +7 = −4

2x +7 = −4−7 −7

2x = −11

x = −11

2

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4x− 12 + x = −7x− 6

5x− 7x + 11 = 13− 11x

z − 5 = 5z − 3

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Take it up a notch3(x + 7)− 11 = 5x− 93x + 21− 11 = 5x− 93x + 10 = 5x− 910 = 2x− 919 = 2x

19

2= x

4(x− 3) + x = −7x− 6

1

3x +

1

4=

1

6− x

5.5x + 3x = 51

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5x = 7(x + 1) + x

3x− (x + 7) = x− 5

7x = 5(x− 3)

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*- No solutions or infinite solutions

5x = 3(x− 3) + 2x

5x = 3x− 9 + 2x

5x = 5x− 90 6= −9No solutions

5(2x + 7)− (10x + 5) = 3010x + 35− 10x− 5 = 3035− 5 = 3030 = 30∞ number of solutionsinfinite number of solutions

2(x− 3) = 2x− 6

4(x + 2)− 2(3x + 3) = 10− 2x

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2.3 Application of Percent Math 085 Chapter 2

Percent:

The expression x% represents the fractionx

100or the decimal number x • 0.01

Conversion to percent - multiply by 100%

Examples:0.175 = 17.5%

3

20= 15%

Conversion to decimal - divide by 100%

11.5% = 0.115

151% = 1.51

Conversion to fraction

5.5% =5.5

100=

55

1000=

11

200

Example 1) In 1960, about 392,000 people received a bachelor’s degree, and by 1998 thisnumber had increased to 1,184,000. Find the percent change in the number of bachelor’sdegrees received over this time period.

1, 184, 000− 392, 000

392, 000= 2.02 = 202%

Example 2) Tuition is currently $125 per credit. There are plans to raise tuition by 8%for next year. What will the new tuition be per credit?

New tuition =$125 + $125 • 8% = $125 + $125 • .08 = $125 • 1.08 = 135 per credit

Example 3) In 1998, 6.9 million, or 37.5%, of all government workers were unionized.How many government workers were there in 1998?

Let A = All government workers

37.5%A = 6.9million0.375A = 6.9 million

A =6.9

0.375million = 18.4 million

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Two bank loans, one for $5000 and the other for $3000 cost a total of $550 in interestfor one year. The $5000 loan has an interest rate 3% lower than the interest rate for $3000loan. Find the interest rate for each loan.

TYPE PRINCIPAL RATE INTEREST$5,000 loan $5, 000

$3,000 loan $3, 000

Total

The interest rate for the $5,000 loan is and the interest rate for the $3,000loan is

You have an alloy that contains 20% brass that needs to be mixed with 990 ounces of analloy that contains 60% brass to create a mixture that is 56% brass.

Amount Percent Pure Brass20% brass 20%

60% brass 60%

56% brass 56%

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You need ounces of 20% brass (If needed, round to 1 decimal place - NOCOMMAS).

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2.4 Formulas Math 085 Chapter 2

A formula is an equation that can be used to calculate an unknown quantity by usingknown values of other quantities.

Examples:

A = lw (rectangle) l = 7, w = 12 A =

A =1

2bh (triangle) b = 8, h = 3 A =

A =1

2(b1 + b2)h (trapezoid) b1 = 7, b2 = 5, h = 4 A =

V = πr2h (volume of a right cylinder) r =3

2, h =

11

2V =

Solving Formulas

F =9

5C + 32 Solve for C

K =2(ab− c)

fgSolve for c

1

R=

1

r1

+1

r2

Solve for r1

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P = 2x + 2y Solve for y

D = rt Solve for t

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Solving Formulas

S = 2LW + 2LH + 2WH Solve for W

t

4+

t

5= 1 Solve for t

T = mg + mf Solve for m

1

p+

1

q=

1

fSolve for f

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ab = ac + d Solve for a

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2.5 Problem Solving Math 085 Chapter 2

Logistics of Solving a Story Problem (Application Problem)

How to solve a story problem1) Read it and understand what is being asked.

a) Jot down important facts - in your own words.2) Define a Variable that represents the solution to the question asked.3) Use a table or data format that allows you to gather and organize the data.4) Use the table (data format) to write an equation.5) Solve the equation.6) Answer the question!! (Complete sentence.)

Types of problems:Number problems

Geometry problems

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For the next 3 problems, let x = the number.

The difference of three times a number, and 1, is the same as twice a number. Find thenumber.

3x− 1 = 2x

x = 1

Five times the sum of a number and −1 is the same as 6 times the number. Find thenumber.

5(x + (−1)) = 6x

5x− 5 = 6x

−5 = x

If the difference of a number and four is doubled, the result is1

4less than the number.

Find the number.

2(x− 4) = x− 1

4

2x− 8 = x− 1

48x− 32 = 4x− 14x = 31

x =31

4

A 17-foot piece of string is cut into two pieces so the longer piece is 2 feet longer thantwice the shorter piece. Find the length of both pieces.

x 2x + 2

17 ft

x + (2x + 2) = 173x + 2 = 173x = 15x = 5

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Shorter piece = 5 feet, the longer piece is 12 feet.

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Consecutive Integers Example:1, 2, 322, 23, 24

Consecutive Even Integers Example:4, 6, 832, 34, 36

Consecutive Odd Integers Example:5, 7, 921, 23, 25

First NextInteger Integers Sum of

Three consecutive integers xthe 2nd and 3rd simplified

Three consecutive odd integers xthree consecutive odd integers

A 46-foot piece of rope is cut into three pieces so that the second piece is three times aslong as the first piece and the third piece is two feet more than seven times the length of thefirst piece. Find the lengths of the pieces.

ft

First piece = feet

Second piece = feet

Third piece = feet

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The countries with the most television stations in the world are Russia and China. Russiahas 4066 more television station than China whereas the total stations for both countriesare 10,546. Find the number of television stations for both countries.

China has television stations

Russia has television stations

The code to unlock a student’s combination lock happens to be three consecutive oddintegers whose sum is 51. Find the integers.

The integers are

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Sum of angles of a triangle is 180 degrees.

Examples:

30◦ 90◦

60◦

√3

12

30◦ + 60◦ + 90◦ = 180◦

Two angles are complementary if their sum is 90◦.Two angles are supplementary if their sum is 180◦.

An Isosceles Triangles has two sides of equal length.

An Equilateral Triangle has all sides of equal length.

Area and Perimeter

Perimeter Area

Triangle Add all sides together A =1

2bh

Rectangle P = 2l + 2w A = lw

Trapazoid Add all sides together A =1

2(b1 + b2)h

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In a triangle adding the measure of the two smaller angles and then subtracting 10◦ isequal in measure to the largest angle. The middle angle is one more degree then the smallest.

measure of smallest angle =

measure of middle angle =

measure of largest angle =

The larger angle measures three degrees less than twice the measure of the smaller angle.If ’S’ represents the measure of the smaller angle and these two angles are complementary,find the measure of each angle.

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Find the Area of the figure below.

21 inches

8 inches

11 inches

The Area is square inches

Prior to United States Homeland Security Department taking over immigration, it was alocal CNMI Immigration that allowed for contract workers to come to the CNMI. A contractworker in Saipan by CNMI law was required to have a minimum of 50 square feet of livingspace in their employer provided housing. If a room measure 7 feet by 7 feet 3 inches, doesthis room provide the required living space?

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2.6 More about Problem Solving Math 085 Chapter 2

More problem Solving

Types of problems:Investment problems

Uniform motion problems

Liquid mixture problems

Dry mixture problems

Number-value problems

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A banker invests money in two investments. The first investment returned 6% simpleinterest. The second investment returned 11% simple interest. If the second investment had$330.00 more money than the first and the total interest for both investments were $146.80.

Find the amount invested in each investment.

TYPE Principal Rate Interest6% simple interest

11% simple interest

Total

The banker invest $ at 6% simple interest

The banker invest $ at 11% simple interest

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Distance: d = rt

Find the missing value

1) r = 4 mph d = 12 miles2) t = 3 hours r = 5 mph3) d = 55 miles t = 5 hours

1) r = 4 mph d = 12 miles t = 3 hours2) t = 3 hours r = 5 mph d = 15 miles3) d = 55 miles t = 5 hours r = 11 mph

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A pilot flies a plane at a constant speed for 5 hours and 30 minutes, traveling 715 miles.Find the speed of the plane in miles per hour.

A train is 100 miles west of St. Louis, Missouri, traveling east at 60 miles per hour. Howlong will it take the train to be 410 miles east of St. Louis?

Two cars pass on a straight highway while traveling in opposite directions. One car istraveling 6 miles per hour faster than the other car. After 1.5 hours the two cars are 171miles apart. Find the speed of each car.

TYPE Distance Rate Timecar 1

car 2

Total

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Car 1 is traveling at miles per hour

Car 2 is traveling at miles per hour

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If a sports car is traveling at 100 miles per hour, how long would it take the sports car toovertake a family car traveling at 45 miles per hour that left 4 hours earlier? (Round youranswer to the nearest tenth of an hour.)

Let t=

TYPE Distance Rate Timesports car

family car

It will take the sports car hours to overtake a family car.

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Example) A chemist has 20% and 40% solutions of acid available. How many liters ofeach solution should be mixed to obtain 110 liters of 22% acid solution?

Let x =

TYPE AMOUNT CONCENTRATION PURE ACID20% solution

40% solution

22% mixture

You need liters of 20% solutions.

You need liters of 40% solutions.

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A solution contains 15% hydrochloric acid. How much water should be added to 50milliliters of this solution to dilute it to a 2% solution?

Let x =

TYPE AMOUNT CONCENTRATION TOTAL15% solution

0% Water

2% solution

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Currently the two fastest trains are the Japanese Maglev and the French TGV. The sumof their fastest speeds is 718.2 miles per hour. If the speed of the Maglev is 3.8 miles perhour faster the speed of the TGV, find the speed of each

Maglev = mph

TGV = mph

A cashier at a store has $5 and $50-dollar bills. If there are four times as many $5 billsas $50 bills and the total amount of money is $1,610.00. Find the number of each bill.

The cashier has 5-dollar bills and 50-dollar bills.

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Beetles have the greatest number of different species. There are thirty-six times thenumber of beetle species as grasshopper species, and the total number of species for both is407,000. Find the number of species for each type of insect.

beetles =

grasshopper =

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2.7 Solving Inequalitites Math 085 Chapter 2

A linear equality represents a boundary of a linear inequality.

Example:If I am driving the speed limit of 65 MPH I can go any speed less than* or equal to 65 MPH,and be legal. However, if I drive over 65 I will be breaking the law.

* - depending on the state, it can be illegal to drive too slow.

Any linear equality can be converted into an inequality by replacing the = with aninequality symbol.

Graphing( or ) or ◦ → exclude endpoint − endpoint is open[ or ] or • → include endpoint − endpoint is closed

Example:

x < 3-4 -3 -2 -1 0 1 2 3 4

x ≥ −2-4 -3 -2 -1 0 1 2 3 4

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Graph the following on a number line

x ≥ 3-4 -3 -2 -1 0 1 2 3 4

x < 4-4 -3 -2 -1 0 1 2 3 4

x ≤ 0-4 -3 -2 -1 0 1 2 3 4

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Set Builder Notation

{x|x ∈ R, }

{ x | x ∈ R }The set x such that x is a real number (conditions go here)

http://www.purplemath.com/modules/setnotn.htm

Other Types of Numbers Notation.

N the counting numbersZ the integersQ the rational numbers (fractions)R the real numbersI the irrational numbersC the complex numbers

So for the 3 previous examples

{x|x ∈ R, x ≥ 3}

{x|x ∈ R, x < 4}

{x|x ∈ R, x ≤ 0}

When using interval notation, the symbol:

( or ) means exclude endpoint that is "open"[ or ] means include endpoint that is "closed"

Concepts of −∞ and∞ (the concepts of the smallest ’number’ and the largest ’number’)

x ≥ 3 x < 6 x ≤ −1{x|x ∈ R, x ≥ 3} {x|x ∈ R, x < 6} {x|x ∈ R, x ≤ −1}[3,∞) (−∞, 6) (−∞,−1]

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Inequality Set Builder Interval Notationx < 5

↑Not Included

-5 -4 -3 -2 -1 0 1 2 3 4 5

x ≤ 5↑

Included

-5 -4 -3 -2 -1 0 1 2 3 4 5

y ≥ 2↑

Included

-4 -3 -2 -1 0 1 2 3 4

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Just like equality we have properties of inequalities to simplify them

Addition property of Inequalities

Let a, b, c ∈ R

The inequalities:a < b and a + c < b + c are equivalent.a ≤ b and a + c ≤ b + c are equivalent.a > b and a + c > b + c are equivalent.a ≥ b and a + c ≥ b + c are equivalent.

Multiplication property of Inequalities

Let a, b, c ∈ R where c 6= 0

If c > 0 than the inequalities:a < b and ac < bc are equivalent.a ≤ b and ac ≤ bc are equivalent.a > b and ac > bc are equivalent.a ≥ b and ac ≥ bc are equivalent.

If c < 0 than the inequalities:a < b and ac > bc are equivalent.a ≤ b and ac ≥ bc are equivalent.a > b and ac < bc are equivalent.a ≥ b and ac ≤ bc are equivalent.

Algebraically solve then write the answer in set builder, interval notation, and graphi-cally.

x + 6 < 3

Inequality

Set Builder

Interval Notation

graphically

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10 ≥ −1

7y

Inequality

Set Builder

Interval Notation

graphically

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4 + 5x ≤ 9

Inequality

Set Builder

Interval Notation

graphically

5(x + 2) > −2(x− 3)

Inequality

Set Builder

Interval Notation

graphically-4/7 -3/7 -2/7 -1/7 0 1/7 2/7 3/7 4/7

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Algebraically solve then write the answer in set builder, interval notation, and graphi-cally.

5x

8− 3x

4≤ 8

Inequality

Set Builder

Interval Notation

graphically

A rectangle is twice as long as it is wide. If the rectangle is to have a perimeter of atleast 36 inches, what values for the width are possible?

A student scores 65 and 82 on two different 100-point tests. If the maximum score onthe next test is also 100 points, what score does the student need to maintain at least anaverage of 70?

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The cost to produce one tablet is $190 plus a one-time fixed cost of $50,000 for researchand development. The revenue received from selling one tablet is $320.

a) Write a formula that gives the cost C of producing x tablets

C =

b) Write a formula that gives the revenue R from selling x tablets.

R =

c) Profit equals revenue minus cost. Write a formula that calculates the profit P fromselling x tablet computers.

P = R− C

d) At least how many tablets must be sold to have at least a 1,090 profit?

−10 ≤ −9x + 9 < −9

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8.4 Solving Compound Inequalities Math 085 Chapter 2

Interval:http://www.coolmath.com/algebra/cruncher/algebra-problems-interval-notation-2.htm

Interval Notation

Recall:a < b←→ a + c < b + c

a > b←→ a + c > b + c

If c > 0 thena < b←→ ac < bc

a > b←→ ac > bc

If c < 0 thena < b←→ ac > bc

a > b←→ ac < bc

Open interval - end points are not included.

(a, b) = {x|x ∈ R, a < x < b} a b (bounded interval)

Close interval - end points are included.

[a, b] = {x|x ∈ R, a ≤ x ≤ b} a b (bounded interval)

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ExamplesInequality Set Builder Interval Notation Graphically

0 5x < 5 {x|x ∈ R, x < 5} (−∞, 5)

0 5x ≤ 5 {x|x ∈ R, x ≤ 5} (−∞, 5]

2 5y ≥ 2 {y|y ∈ R, y ≥ 2} [2,∞)

The examples in the table are unbounded intervals (because of the ±∞).

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For all of the following leave answers as terminating decimals or fractions.

Example 1) Solve 9 < −3x

9

−3> x

x < −3

The set builder answer is {x|x ∈ R, x < −3}

The interval notation answer is (−∞,−3)

Example 2) Solve 6x + 7 < 11x− 4

7 < 5x− 411 < 5x

11

5< x

The set builder answer is {x|x ∈ R,11

5< x}

The interval notation answer is(

11

5,∞

)


Solve 7 < −2x

The set builder answer is {x|x ∈ R, }

The interval notation answer is

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Solve 5x + 7 < 8x− 9



Solve 4m + 7 ≥ 14(m− 3)



Solve 2(4 + 2x) > 2x + 3(2− 5x)



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1 dimension inequalities

0 2 10{x|x ∈ R, 2 < x} (2,∞)

0 2 10{x|x ∈ R, x ≤ 10} (−∞, 10]

0 2 10{x|x ∈ R, 2 < x ≤ 10} (2, 10]

Recall 2 < x ≤ 10 is shorthand for 2 < x and x ≤ 10.

Solve −3 < t + 1 ≤ 5

−3 < t + 1 ≤ 5 What you do to 1 part you do to ALL!−1 −1 −1−4 < t ≤ 4

The set builder answer is {x|x ∈ R,−4 < x ≤ 4}

The interval notation answer is (−4, 4]

1 < x + 2 and x− 3 < 4 Work each piece individually!−1 < x and x < 7−1 < x < 7

The set builder answer is {x|x ∈ R,−1 < x < 7}

The interval notation answer is (−1, 7)

Careful: −3 < x and x > 10 CANNOT be written as −3 < x > 10 because the senseof the inequalities are different!

−3 < x and x > 10 should be written as −3 < x and 10 < x

−3 0 10

Pick a number between −3 and 10. I’ll use 0. Since this is an ’and’ problem both mustbe true at the same time, since −3 < 0 is true and 10 < 0 is false. We conclude that 10 < x

is the correct inequality.

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Now consider −3 < x or x > 10

Rewrite the inequalities so they have the same sense −3 < x or 10 < x

−3 0 10

Pick a number between −3 and 10. I’ll use 0. Since this is an ’or’ problem one or bothmust be true at the same time, since −3 < 0 is true and 10 < 0 is false. We conclude that−3 < x is the correct inequality.

Example 1) Solve 8− x < 0 and 2x + 1 > 9

8 < x and 2x > 88 < x and x > 48 < x and 4 < x

Use 5 as a test number and conclude that 8 < x is the correct inequality.

The set builder answer is {x|x ∈ R, 8 < x}

The interval notation answer is (8,∞)

Example 2) Solve 3− 2x ≤ 11 or 2x + 3 > 9

The set builder answer is {x|x ∈ R,−4 ≤ x}

The interval notation answer is [−4,∞)

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Solve 1− x < −2 and 2x + 1 > 9



Solve 5− 3x ≤ 8 or 2x + 1 > 7



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Solve the following. Write answers in inequality form

1) If f(x) = 2x− 7 and g(x) = 5x− 9 find all values of x for which f(x) < g(x).

2) The Bayside Inn offers two plans for parties. Under plan A, the inn charges $40 foreach person in attendance. Under plan B, the inn charges $1,510 plus $25 for each person inexcess of the first 25 who attend. For what size parties will plan B cost less? (Assume thatmore than 25 guests will attend.)

Let G = number of guest attending the party

Plan A Plan BCost

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3) The function C(F ) =5

9(F − 32) can be used to find the Celsius temperature C(F )

that corresponds to F Fahrenheit.

a) Gold is solid at Celsius temperatures less than 1063◦C. Find the Fahrenheit tempera-tures for which gold is solid (leave answer as a decimal).

b) Silver is solid at Celsius temperatures less than 960.8◦C. Find the Fahrenheit temper-atures for which silver is solid (leave answer as a decimal).

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2.1 Solving Equations Using Properties of Equality Math ... · 2.1 Solving Equations Using Properties of Equality Math 085 Chapter 2 ... 2.5 Problem Solving ... 2.1 Solving Equations