Pre-AlgebraUnit 3

Solving Equations/Intro to Functions

Section 1Properties and Operations

Commutative PropertyCommutative Property of Addition:In a sum, you can add the numbers in any order.

4+(-7)=

Commutative Property of Multiplication:In a product, you can multiply the numbers in any order.

8(-5)=

Associative PropertyAssociative Property of Addition:Changing the grouping of the numbers in a sum does not change the sum.

(9+6)+2=

1

Associative Property of Multiplication:Changing the grouping of the numbers in a product does not change the product.

(3∙10)∙4=

Example 1.1You buy a portable CD player for $48, rechargeable batteries with charger for $25, and a CD case for $12. Find the total cost.

Example 1.2Evaluate 4xy when x=-7 and y=25

Example 1.3(17+36)+13

2

8(-3)(5)

3.4+9.7+7.6

Evaluate 5x2y when x=-6 and y=20

Simplify a Variable ExpressionTo simplify a variable expression, follow the same rules of addition and multiplication as with numbers.

x+3+6

4(8y)

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Example 1.4m+5+9

6(3k)

4+x+(-1)

(2r)(-5)

Identity PropertiesIdentity Property of Addition:The sum of a number and the additive identity, 0, is the number.

-6+0=

Identity Property of Multiplication:The product of a number and the multiplicative identity, 1, is the number.

4∙1=

4

Example 1.5Identify the Propertya. (-5)(1)=-5

b. 2+(-9)=-9+2

c. y2+0=y2

d. 2(pq)=(2p)q

Unit AnalysisUnit Analysis: a method to convert ________________________

_________________________.

Use unit analysis to find a conversion factor that converts a given measurement to different units.

Example 1.6The Steel Dragon 2000 is the world's longest roller coaster. Its length is 2711 yards. How long is the roller coaster in feet?

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Example 1.7This week, you rode in a car for 42 miles, rode a bike for 5 miles, and rode in a bus for 23 miles. Find the total distance.

Example 1.8Simplify:x+5+2

3(9y)

n+6+7

(4r)(-3)

Example 1.9Identify the property Illustrated:(3+2)+4=3+(2+4)

0+b=b

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1(-7)=-7

cd=dc

(2∙6)∙3=2∙(6∙3)

q+(-r)=-r+q

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Section 2The Distributive Property

Problem: You and a friend are going on a camping trip. You each buy a backpack that costs 490 and a sleeping bag that costs $60. What is the total cost of the camping equipment?

Method One: Find the cost of one backpack and one sleeping bag. Then multiply the result by 2, the number of each item bought

Method Two: Find the cost of two backpacks and the cost of two sleeping bags. Then add the costs.

Equivalent Numerical Expressions: expressions with the

____________________

In the camping example, 2(90+60) and 2(90)+2(60) are equivalent numerical expressions

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The Distributive PropertyUsed for evaluating the product of a number and a sum or difference.

4(6+3)=

(6+3)4=

5(7-2)=

(7-2)5=

Example 2.1Use the distributive property to write an equivalent expression. Then evaluate the expression.3(8+5)

(2+9)2

6(11-4)

(3-14)(-5)

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Equivalent Variable Expressions:two variable expressions that have the same value for all values of the variable

Example 2.2Write an equivalent variable expression:3(x+7)

(n+4)(-2)

-5(2y-3)

Example 2.3Write an equivalent variable expression:8(x+2)

(7-t)(-4)

9(3m+5)

-2(6y-4)

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Example 2.4

Example 2.5

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Section 3Simplifying Variable Expressions

Term:

Coefficient:

Constant term:

Like terms:

For example: 5x+4x+7Terms:

Coefficients:

Constant terms:

Like terms:

Steps to Identify Parts of an Expression1. Rewrite the expression as a sum.

Change subtraction to _____________

2. Identify the parts of the expression.

List parts and their signs! (+/-)

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Example 3.1Identify the terms, like terms, coefficients, and constant terms of the expression y+8-5y-3

1. Write the expression as a sum.

2. Identify the parts of the expression.Terms:

Like Terms:

Coefficients:

Constant Terms:

What about the y? What is the coefficient of y? Think about the identity property!

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Example 3.2Identify the terms, like terms, coefficients, and constant terms of the expression 5-2x-3+x

1. Write the expression as a sum.

2. Identify the part of the expression.* Note that because x=____, the coefficient of x is ____.

Terms:

Like terms:

Coefficients:

Constant terms:

Example 3.3Identify the terms, like terms, coefficients, and constant terms of the expressions.1. 4y-6+3y

2. 9+w-5-8w

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Example 3.4Identify the terms, like terms, coefficients, and constant terms of the expressions.1. 3x+2+5x

2. -7b+3+b-10

3. 5+8w-6-w

Example 3.5Simplify and then identify the terms, like terms, coefficients, and constant terms of the expressions.

1. 2(x-4)+9x+1

2. 3k-8(k+2)

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3. 4a-(4a-3)

Simplifying ExpressionsUse the distributive property backwards to write an expression as a single term.

7x+4x=(7+4)x=11x

Like terms: Combined to simplify the expression

A variable expression is simplified if it contains no grouping symbols and all like terms are combined.

Steps to Simplify an Expression1. Rewrite the expression as a _________.

2. Use the ________________________ (if necessary) to remove

parentheses from the expression.

3. Use the ________________________ (if necessary) to place

like terms next to each other in the expression.

4. Use the ________________________ to group like terms

together.

5. Use the ________________________ in reverse to combine like

terms.

6. __________________ (if necessary).

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Example 3.6Simplify:4n-7-n+9

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

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Example 3.7Simplify using the six steps:a. 3x+2+5x

b. -7b+3+b-10

c. 5+8w-6-w

Shortcut!A quick way to combine like terms containing variables is to add their coefficients mentally. For example to add 4n-n=4n+(-n), add the coefficients 4+(-1)=3. This means that 4n+(-n)=3n.

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Example 3.8Simplify:2(x-4)+9x+1

3k-8(k+2)

4a-(4a-3)

Example 3.9Simplify:4(x+1)+2x+5

10y-3(6-y)

8c+2-(c+2)

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Example 3.10During your workout, you spend a total of 45 minutes jogging and swimming. You burn 14 calories per minute when jogging and 8 calories per minute when swimming.

a. Let j be the time you jog (in minutes). Write an expression in terms of j for the total calories you burn during your workout.

b. Find the total number of calories burned if you jog for 20 minutes.

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Section 4Variables and Equations

Equation:

Solution:

Example of an equation:

Example 4.1Write the verbal sentence as an equation:1. The sum of x and 6 is 9.

2. The difference of 12 and y is 15.

3. The product of -4 and p is 32.

4. The quotient of n and 2 is 9.

Example 4.2Tell whether 9 or 7 is a solution of x-5=2.

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Example 4.3Write the verbal sentence as an equation:1. The sum of 3 and z is -10.

2. The quotient of m and 6 is 4.

Tell whether -5 or 5 is a solution of -8y=40.

Example 4.4Write the verbal sentences as equations:1. The sum of x and 4 is 8.

2. The difference of 7 and y is 13.

3. The product of -2 and p is 24.

4. The quotient of n and 3 is 5.

5. The sum of x and 7 is 12.

6. The quotient of n and 4 is 16.

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7. Tell whether 7 or 8 is a solution of x-3=5.

8. Tell whether 8 or 10 is a solution of x-4=6.

Solving EquationsFinding _________________________________ is called solving the equation. You can use mental math to solve a simple equation by thinking of the equation as a question.x+3=11

What number plus 3 equals 11?Solution:

Check:

Example 4.516-m=9

Question:

Solution:

Check:

20=5tQuestion:

Solution:

Check:

23

y6 =−3

Question:

Solution:

Check:

Example 4.6x-10=7

Question:

Solution:

Check:

2+n=-6Question:

Solution:

Check:

3w=-15Question:

Solution:

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Check:

4=36s

Question:

Solution:

Check:

Example 4.7From 1998 to 2002, biologist Jane Shen-Miller grew several lotus plants from ancient seeds she found in China. The oldest seed was about 500 years old. Estimate the year when this seed was formed.Step 1: Write a verbal model.

Step 2: Write a variable equation (remember we are ESTIMATING).

Step 3: Use mental math to solve the equation.

Step 4: Answer the question.

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Example 4.8Go-cart rides cost $5 each at a county fair. During the first day of the fair, the go-cart operator takes in a total of $1000. How many times did people ride the go-carts that day? Write and solve an equation to find the answer.

Example 4.9Use mental math to solve the equations.1. x+4=7

2. 12-n=5

3. 18=3t

4. y4 =−5

5. x-8=10

6. 24=4m

7. c3=9

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Section 5Solving Equations Using Addition and Subtraction

Inverse Operations: Two operations that undo each otherExample:

Equivalent Equations: Equations with the same solution

Subtraction Property of EqualitySubtracting the same number from each side of an equation produces an equivalent equation.

Example 5.1Solve: x+9=-3 and then check the answer.

Addition Property of EqualityAdding the same number from each side of an equation produces an equivalent equation.

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Example 5.2Solve: 23=y-11 and check your solution.

Example 5.3Solve showing all steps and check your solution.1. x+8=19

2. -7=y+13

3. n-4=-11

4. 26=p-61

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Example 5.4Solve showing all steps and check your solution.1. x+5=-2

2. 12=y-7

3. x+6=19

4. -5=y+12

5. m-3=-11

Example 5.5You weigh a horse by putting it in a trailer of known weight and weighing the horse and trailer together on a truck scale. The weight of the trailer alone is 2150 pounds. The combined weight of the horse and trailer is 3375 pounds. Write an equation to model the situation. What is the weight of the horse?

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Example 5.6While holding his cat, Ben steps on a scale. The scale reads 161 pounds. Ben weighs 148 pounds. What is the weight of the cat?

Example 5.72+x=24

x-(-9)=-5

m+4-9=12

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Section 6Solving Equations Using Multiplication or Division

Division Property of EqualityDividing each side of an equation by the same nonzero number produces an equivalent equation.

Example 6.1Solve -6x=48 and check your solution.

Multiplication Property of EqualityMultiplying each side of an equation by the same nonzero number produces an equivalent equation.

Example 6.2Solve 9=w/7 and check your solution.

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Example 6.3Solve showing all steps and check your solution.1. 2x=18

2. -60=-5a

3. y/2=13

4. -8=b/8

Example 6.4Solve showing all steps and check your solution.1. 5x=45

2. -56=-8y

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3. 5=w/11

4. m/4=11

5. -9=c/6

6. -7x=42

Example 6.5On January 4, 2003, the SOHO spacecraft, which studies the sun, including high-speed gas called solar wind that is ejected from the sun and travels throughout the solar system, detected solar wind traveling at about 2 million kilometers per hour. The sun is about 150 million kilometers from Earth. About how long did it take the solar wind to reach Earth?

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Section 7Decimal Operations and Equations With Decimals

Skill Check1. 2 + (-7)

2. -4 - (-9)

3. -10 + (-3)

4. 8 - (-12)

Name decimals that lie between the integers.

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Rules For Decimal Operations1. Addition

a. Both positive: Just addb. Both negative: Add the absolute values of the numbers. The

sum should then be negative.c. Opposite signs: Subtract the absolute value of the smaller

number from the absolute value of the larger. The sign will be the sign of the larger number.2. Subtraction: Change to adding the opposite and use the rules for addition.3. Multiplication

a. Both positive: Just multiply. The product is positive.b. Both negative: Multiply the two numbers. The product is

positive.c. Opposite signs: Multiply the two numbers. The product is

negative.4. Division: Follow the same rules as multiplication.

Example 7.11. -2.9 + (-6.5)

2. -25.38 - (-42.734)

Example 7.23. -1.3 + (-4.2)

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4. 10.57 + (-6.89)

5. 9.817 - (-1.49)

Example 7.36. -0.7(18.4)

7. -4.5(-9.25)

8. -29.07 ÷ (-1.9)

9. 16.83 ÷ (-3.3)

Example 7.410. 3.1(-6.8)

11. -11.41 ÷ (-0.7)

12. -15.841 ÷ 2.17

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Example 7.513. -1.7 + (-3.4)

14. -21.29 - (-34.62)

15. -2.8 + (-5.9)

16. 7.12 - (-3.46)

Example 7.617. -0.4(13.7)

18. -2.5(-6.75)

19. -23.49 ÷ (-2.9)

20. 18.05 ÷ (-1.9)

Example 7.721. -2.8(-5.9)

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22. 7.093 ÷ (-3.46)

Challenge Example 11. Simplify: 8.9(3.8m-6.7)

2. Simplify: 7(2.4r+6.3)-4.24

3. Simplify: -2.7(4-3.1a)+14.5a

4. Simplify: 10.5+2(6.7-4.2x)+7.5x

Challenge Example 2Find the mean, median, mode, and range:5. 7.5, -2.3, -5.7, 6.6, -0.2, 9.7

6. 12.3, 8.5, -6.8, -0.5, 2.1, 3.0

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Solving Equations Involving Decimals

Example 7.81. x+4.7=3.5

2. y-6.91=-2.26

Example 7.93. x+3.8=5.2

4. a+10.4=-1.17

5. 6.29+c=4.01

Example 7.106. y-7.8=22.3

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7. r-0.88=-0.56

8. -9.34=t-2.75

Example 7.119. -0.6m=-5.1

10. n/(-8)=1.75

Example 7.1211. 7x=40.6

12. -1.8u=6.3

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13. y/11.5=0.4

14. -9.1=v/(-5.9)

Example 7.13When a chipmunk hibernates, its weight decreases by about 0.31 pound. After hibernation, a chipmunk weighs about 0.35 pound. Find the weight of a chipmunk before hibernation.

Example 7.14You use an automated teller machine (ATM) to deposit a check for $122.94 into your savings account. Your receipt from the ATM shows a balance of $286.59 after the deposit. Find the balance of your savings account before the deposit.

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Example 7.1515. x+6.3=4.8

16. y-5.74=-3.51

17. x+5.6=9.4

18. -3.5=y+1.2

19. m-5.3=-7.2

Example 7.1620. 0.8m=4.8

21. n/5=-2.15

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22. 6x=-43.2

23. y/(-3.1)=-8.4

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Section 8Solving Two-Step Equations

To “UNDO” an EquationOrder of Operations in Reverse:

S

A

D

M

E

P

Example 8.1Solve and Check:3x+7=-5

4x+1=5

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Example 8.23n+8=2

1=2r+9

2=6h+20

Example 8.3x/2-3=1

b/4-8=1

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c/6-2=6

2=d/5-1

Example 8.47-4y=19

12-4s=-12

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6-2m=8

-2=5-n

Example 8.5You are buying a drum set that costs $495. The music store lets you make a down payment. You can pay the remaining cost in three equal monthly payments with no interest charged. You make a down payment of $150. How much is each monthly payment?

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Section 9Solving Equations Having Like Terms and Parenthesis

Example 9.1Your school's basketball team is playing in the championship game. For the game, the poms want to buy a banner that costs $47. They also want to buy small items to give to students in the stands. Pompoms cost $5.20 each. Noisemakers cost $.80 each. The cheerleaders have a total budget of $377 for the game. If they buy equal numbers of pompoms and noisemakers, how many can they afford to buy?

Example 9.2-21=7(3-x)

-3(8-4x)=12

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5x-2(x-1)=8

Example 9.32n-40+2n=15

2(s-1)=6

13=2y-3(y+4)

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Example 9.413t-7-10t=2

22+4y-14=0

2d+24+3d=84

Example 9.54(x+5)=16

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3(7-2y)=9

-2(z+11)=6

Example 9.6-5(3n+5)=20

-30=6(f-5)

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12=3(m-17)

Example 9.7A family of 5 people has $200 to spend on fishing rods and fishing licenses. They spend a total of $20 on licenses. Assuming they buy 5 identical rods, what is the maximum amount they can spend on each rod?

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Section 10Solving Equations with Variables on Both Sides

Step One: Get all variable terms on one side of the equation by adding or subtracting.Step Two: Solve for the variable

Example 10.1Solve:7n-5=10n+13

Example 10.25n-2=3n+6

8y+4=11y-17

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m-1=9m+15

Example 10.3The Spanish club is arranging a trip to a Mexican restaurant in a nearby city. Those who go must share the $60 cost of using a school bus for the trip. The restaurant's buffet costs $5 per person. How many students must sign up for this trip in order to limit the cost to $10 per student?

**When you solve an equation, you may find that it has no solution or that every number is a solution.-- No solution when you solve and get a false statement: Such as 3=9-- Every number is a solution if you get a statement that is always true: Such as x=x or 2=2

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Example 10.35(2x+1)=10x

6x+2=2(3x+1)

Example 10.4Find the perimeter of the square (no x's involved).

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Section 11Solving Inequalities Using Addition and Subtraction

x<3

y>2

z≤4

n≥2

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Example 11.1The freezing point of water is 0o C. At temperatures at or below the freezing point, water is a solid (ice). Write an inequality that gives the temperatures at which water is a solid. Then graph the inequality.

Addition and Subtraction Properties of InequalityAdding or subtracting the same number on each side of an inequality produces an equivalent inequality.

**Note this is exactly the same as when solving equations!

Example 11.2

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m+5≥10

-10>x-12

Example 11.3n+7>3

10≥y+4

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z-5<1

Example 11.4You are competing in a triathlon, a sports competition with three events. Last year, you finished the triathlon in 85 minutes. The table shows your times for this year's first two events. What possible times can you post in the running event and still beat last year's finishing time?

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Section 12Solving Inequalities Involving Multiplication and Division

Multiplication and Division for solving inequalities follows basically the same rules as solving equations, with one MAJOR exception.

If c is negative:

**When multiplying or dividing by a negative, you switch the inequality symbol**

Example 12.19.14m/(-3)>3

-10t≥34

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Example 12.2n/6>7

t/(-4)≤8

2x>-8

-7s≤14

Example 12.3Some flocks of Canada geese can fly nonstop for up to 16 hours. In this time, a flock can migrate as far as 848 miles. At what average speeds can such a flock fly during migration?

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Section 13Solving Multi-Step Inequalities

Review: Steps to Solve EquationsStep 1: Distribute if necessaryStep 2: Combine like terms on the same side of the equal signStep 3: Get all variable terms on one side of the equation using addition or subtractionStep 4: Begin to isolate the variable using addition and subtractionStep 5: Isolate the variable using multiplication and divisionStep 6: Check

Steps to Solve an InequalityFollow EXACT same steps as to solve an equation!The inequality sign must face the same direction in each step unless...?

To check: Pick a number that falls in the solution and plug it in!

Example 13.1x÷(-4)-6≥-5

4x+1>1

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Example 13.210>6+y÷5

5y+2≤y+34

Example 13.36+y≥2y-3

19≥x÷90-25

Example 13.43m-8>-30+5m

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-3x-3<2x-83

Example 13.5Your school's soccer team is trying to break the school record for goals scored in one season. Your team has already scored 88 goals this season. The record is 138 goals. With 10 games remaining on the schedule, how many goals, on average, does your team need to score per game to break the record?

Example 13.6You plan to go ice skating often this winter. The skating rink charges $4 for admission. You can either rent ice skates at the skating rind for $5 per day or buy your own pair for $45. How many times do you have to use the ice skates in order for the cost of buying them to be less than the total cost of renting them?

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Section 14Using Multiplicative Inverses to Solve Equations

Multiplicative InversesThe product of a number and its multiplicative inverse is ______.

The multiplicative inverse of a number is its _________________.

ReciprocalWhen did we learn about this word?

What is it?

Multiplicative InverseGiven a rational number _______, where a≠0 and b≠0, the

multiplicative inverse is _______.

So _______________

With numbers this is:

The multiplicative inverse of _______ is ________.

So _______________

Negative Numbers

Given the fraction ________, what would you need to multiply by

to get 1?

So, what is the rule for the multiplicative inverse of a negative rational number?

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Example 14.1Find the multiplicative inverse and multiply to check your answer:a. 2/9

b. 3/8

c. -5/12

d. -1/6

Mixed NumbersHow do you think we will find the multiplicative inverse of 2 1

2?

What is the rule for finding the multiplicative inverse of a mixed number?

Example 14.2Find the multiplicative inverse of the number and multiply to check your answer:

a. 2 34

b. −1 57

c. 3 516

d. −8 12

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Example 14.3Solve and Check:3x+7=-5

4x+1=5

Example 14.4x/2-3=1

b/4-8=1

c/6-2=6

2=d/5-1

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Example 14.513t-7-10t=2

22+4y-14=0

2d+24+3d=84

Example 14.64(x+5)=16

3(7-2y)=9

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-2(z+11)=6

Example 14.75n-2=3n+6

8y+4=11y-17

m-1=9m+15

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Example 14.85(2x+1)=10x

6x+2=2(3x+1)

One Way to Look at It4/7x=-12

Much Shorter MethodWhat is the multiplicative inverse of 4/7?

Multiply this to each side!

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Example 14.9 5 1

2+ p=6

−34b=2

Example 14.10 x−3=−5 1

2

2 110n=1 1

6

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Example 14.11 25x+1

5=3

4

37−1

4y=1

2

Example 14.12 −8

15x− 7

15=2

5

72

38= 5

12− 11

24x

Example 14.13The attendance for the first night of a school play is 1/4 of the total attendance. The attendance for the second night is 30 more than 1/5 of the total attendance. The attendance for the third night is 2/7 of the total attendance. The last night has an attendance of 340 people. Find the total attendance for the four nights of the play.

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Section 15Equations and Inequalities with Rational Numbers

Clearing FractionsWhat is our method to solve equations with fractions right now? For example, if you were to solve the following problem, what steps would you take? −3

4x+ 3

8=27

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Another method for solving equations involving fractions is to clear the fractions by multiplying each side of the equation by the LCD of the fractions. The resulting equation is equivalent to the original equation.

Example 15.1 −5

6x+1

2=3

4

Example 15.2 12x+ 7

10=4

5

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Example 15.3 38x−2

3= 7

12

Example 15.4 −2

9=3

4x−1

6

Example 15.5 27+3x= 9

21

75

Example 15.6 25x−1

2=27

10

Example 15.7You need 1/6 cup of flour for a recipe but you only have measuring cups for 1/2 cup and 1/3 cup. How can you measure out 1/6 cup of flour?

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Example 15.8You are helping a classmate with his or her homework and you notice that the following problem was completed incorrectly. Find the error and explain what was done wrong.

23x+5=5

2

6( 23x)+5=6( 5

2)

4 x+5=154 x=10

x=104

x=52

Example 15.9The water level in a rain gauge is 2 3

4 inches. A steady rain raises

the water level by 18 inch each hour. When the rain stops, the gauge

reads 4 inches. How many hours did the rain last?

Clearing Decimals

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We have learned how to solve equations involving decimals already. How would we solve the following equation?54.7 = -9.3 n + 8.2

Another method to solve equations involving decimals is to multiply both sides of the equation by 10n, where n is the greatest number of decimal places in any of the terms.

Example 15.102.3 = 5.14 + 0.8 m

Example 15.112.2 b - 3.64 = 8.46

Example 15.125.5 x + 6.6 = 16.5

Example 15.13

78

-0.67 x + 2.278 = 5.695

Example 15.143.12 = -1.248 - 0.78 x

Example 15.152.6 x - 8.32 = 20.80

Example 15.16

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The surface area S of the rectangular solid is S = 2 (21.7) + 2 (3.5 x) + 2 (6.2 x). Find the length x if the surface area is exactly 206.36 square centimeters.

Example 15.176n - 3 ≤ -9

3m - 8 > -30 + 5m

Example 15.18

80

11 - 4z < -1

3 + b/3 < 7

Inequalities with FractionsWe can use the same two methods we used to solve equations with fractions to solve inequalities with fractions. The two methods are:1. Multiplying by the multiplicative inverse (reciprocal)2. Clearing fractions

Example 15.19 −4

5x−1

2< 3

10

Example 15.20

81

−34m−1

8≤−1

4

Example 15.21 58x+ 1

2≤ 13

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Example 15.22 713x−1> 1

2

Example 15.23

82

56n−1

5← 8

15

Example 15.24 17r+ 53

56> 6

7

Example 15.25 −31

4←13+ 7

8x

Example 15.26

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A sign in a clothing store says to take 1/3 off the marked price of a shirt. You have $20 in cash and a $5 gift certificate. What are the original prices of the shirts you can afford to buy?

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