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Solving Equations

08/09/12

Table of Contents

Learning Objectives

Evaluating Expressions

Solving Equations

Solving Inequalities

Practice

1

2

3

4

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Learning Objectives

LO 1 Understand the difference between evaluating and solving

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LO 2 Correctly evaluate expressions

LO 3 Correctly solve equations and inequalities

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Definitions

Definition 1 Evaluate – “Plug in” a value(s) and simplify an expression/equation

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Definition 4 Equations – contain =

Definition 5 Inequalities – contain ≠ < > ≤ ≥

Examples

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Expression -> 2x + 3Equation -> y = 2x + 3Inequality -> y > 2x + 3

Definition 3 Expressions – do not contain = ≠ < > ≤ ≥

Definition 2 Solve – Using reverse operations, find value(s) that fit an equation

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Previous Knowledge

PK 1 Basic Operations and Properties

PK 2 Simplify by Combining Like Terms

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PK 3 Exponent Rules

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Evaluating Expressions

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Step1 Evaluate means you need two thingsAn expression or equationSome value

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Step2 Evaluate means “Plug it in!”

Step3 Simplify

Evaluate 2x + 3 when x = 42x + 3 x = 4

2 (4) + 3 =

8 + 3 = 11

2x + 3 =

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Now you try!

Evaluate 5x + 6 when x = − 4

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Step1 Evaluate means you need two thingsAn expression or equationSome value

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Step2 Evaluate means “Plug it in!”

Step3 Simplify

Evaluate 5x + 6 when x = − 45x + 6 x = − 45 (− 4) + 6 =

− 20 + 6 = − 14

5x + 6 =

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Now you try something harder!

Evaluate 5x2 + 6x + 10 when x = 2

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Step1 Evaluate means you need two thingsAn expression or equationSome value

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Step2 Evaluate means “Plug it in!”

Step3 Simplify

Evaluate 5x2 + 6x + 10 when x = 25x2 + 6x + 10 x = 2

5 (22) + 6(2) + 10 =

5(4) + 12 + 10 =

5x2 + 6x + 10 =

20 + 12 + 10 = 42

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Now you try a really hard one!

Evaluate 4x2 + 5x + 22 x3

when x = - 2

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Step1 Evaluate means you need two thingsAn expression or equationSome value

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Step2 Evaluate means “Plug it in!”

Step3 Simplify

Evaluate 4x2 + 5x + 18 / x3 when x = – 24x2 + 5x + 18 / x3 x = – 2

4(– 22) + 5(– 2) + 18 / (– 2)3 =

[4(4) – 10 + 18] / – 8 =

4x2 + 5x + 18 / x3 =

[16 – 10 + 18]/ – 8 =

24/ – 8 = – 3

Note The only reason the problem is “hard” is it has a – sign!Always scan the problem for – signs before starting a problem

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Solving Equations

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Step1 Solve means LOLLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 I am going to use a teeter totter to demonstrateRewrite the equation (= over the fulcrum)The = means the equation is in “balance” at all times!

Note I will show you what happens when you do not do the opposite!Move any number not attached to a variable(letter).Here I will move the 6.Watch what happens!

Solve 5x + 6 = 46

5x + 6 = 465x = 46+ 6

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Step3 Let’s do it the correct wayLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step4 What is the opposite operation of + 6?Note that the equation is still in balance

Step5 What is the opposite operation of 5x?Note that the operation is still in balance

Solve 5x + 6 = 46

5x + 6 = 465x = 46− 6x = 40/ 5

Step6 The solution is…Until you get really good, check by “Plugging it in”!

x = 8

5(8) + 6 = 4640 + 6 = 46

46 = 46

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Now you try

Solve 4x − 10 = 90

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Step1 Solve means LOLLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 Use a teeter totter for nowRewrite the equation (= over the fulcrum)The = means the equation is in “balance” at all times!

Step3 Move any number not attached to a variable(letter)Use the opposite operationContinue until you solve the problem!

Solve 4x – 10 = 90

4x – 10 = 904x = 90+ 10

Step4 Plug in your answer and check your work!

4x = 100/ 4x = 100 x = 25

4(25) – 10 = 90100 – 10 = 90

90 = 90

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Now you try without the teeter totter!

Solve 2(4x − 3) = 3x + 15

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Step1 Solve means LOLLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 No need for the teeter totter!The = means the equation is in “balance” at all times!

Step3 DistributeMove like terms together (LOL)Use the opposite operationContinue until you solve the problem!

Solve 2(4x – 3) = x + 15

Step4 Check your answer

8x – 6 = x + 158x = x + 15 + 68x = x + 21

8x – x = 217x = 21 x = 21/7

x = 3

2[(4(3) – 3] = 3 + 15

2[12 – 3] = 3 + 15

2[9] = 3 + 15

18 = 3 + 15

18 = 18

+ 6

– x

/7

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Now you try!

Solve (14x − 3)/5 = 4x + 15

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Step1 Solve means LOLLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 No need for the teeter totter!The = means the equation is in “balance” at all times!

Step3 Distribute if/when neededMove like terms together (LOL)Use the opposite operationContinue until you solve the problem!

Solve (14x – 3)/5 = 4x + 15

Step4 Check your answer

14x – 3 = 5(4x + 15)14x – 3 = 20x + 7514x = 20x + 75 + 3 14x = 20x + 78

14x – 20x = 78 – 6x = 78

x = – 13

[14(-13) – 3]/5 = 4(-13) + 15

[– 182 – 3]/5 = – 52 + 15

– 185 / 5 = – 52 + 15

– 37 = – 52 + 15+ 3

– 20x

/ – 6

5( )

x = 78/ – 6

– 37 = – 37

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Solving Inequalities

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Step1 Inequalities are solved like equationsLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 The only difference is when you divide or multiply by a negativeFlip the symbol around!

Step3 Distribute if/when neededMove like terms together (LOL)Use the opposite operationContinue until you solve the problem!

Solve [14x – 3]/5 ≤ 4x + 15

Step4 Check your answerChoose an easy number to work with; here 0 > -13 so we will use 0

14x – 3 ≤ 5[4x + 15]14x – 3 ≤ 5 4x + 5 15 ∗ ∗14x – 3 ≤ 20x + 75

14x ≤ 20x + 75 + 314x – 20x ≤ 78

– 6x ≤ 78

x ≥ – 13

[14(0) – 3]/5 ≤4(0)+ 15

(– 3)/5 ≤15

– 0.6 ≤ 15

This is True

The answer is correct

x ≥ 78/– 6

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Now you try!

Solve − 2[3x − 4] ≥ 4x − 12

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Step1 Inequalities are solved like equationsLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 The only difference is when you divide or multiply by a negativeFlip the symbol around!

Step3 Distribute if/when neededMove like terms together (LOL)Use the opposite operationContinue until you solve the problem!

Solve – 2 [3x – 4] ≥ 4x – 12

Step4 Check your answerChoose an easy number to work with; here 0 < 2 so we will use 0

– 2 3x ∗ – 2(– 4) ≥ 4x – 12– 6x + 8 ≥ 4x – 12

– 6x ≥ 4x – 12 – 8 – 6x ≥ 4x – 20

– 6x – 4x ≥ – 20 – 10x ≥ – 20

x ≤ 2

– 2[3(0) – 4] ≥ 4(0) – 12

– 2(– 4) ≥ – 12

8 ≥ - 12

This is True

The answer is correct

x ≤ – 20 /– 10

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Now you try one last one!

Solve 2[4x − 5] ≥ 4x − 12

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Step1 Inequalities are solved like equationsLOL means Letter on LeftEverything else goes to the right by using reverse operations!

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Step2 The only difference is when you divide or multiply by a negativeFlip the symbol around!

Step3 Distribute if/when neededMove like terms together (LOL)Use the opposite operationContinue until you solve the problem!

Solve 2 [4x – 5] ≥ 4x – 12

Step4 Check your answerChoose an easy number to work with; here 0 ≥ - 1/2 so we will use 0

2 4x ∗ + 2(– 5) ≥ 4x – 128x – 10 ≥ 4x – 12

8x ≥ 4x – 12 + 10 8x ≥ 4x – 2

8x – 4x ≥ – 2 4x ≥ – 2

x ≥ - ½

2[4(0) – 5] ≥ 4(0) – 12

2(– 5) ≥ – 12

– 10 ≥ – 12

This is True

The answer is correct

x ≥ – 2/4

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Practice

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Problem Answer

Evaluate 3x – 7 when x = 8

Evaluate 3x2 when x = 5

Evaluate – 3x2 when x = -10

Evaluate – 4x2 + 5x – 10 when x = 5

Solve 4x – 10 = 90

Solve 2(x – 10) = 80

Solve 4(3x + 30)/5 = 120

Solve 4x + 3 ≤ 31

Solve – 2(3x – 15) > – 60

> 17

>

>

>

>

>

75

– 300

– 85

x = 25

x = 50

> x = 40

> x ≤ 7

> x < 15

clear answers