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Solving Equations
08/09/12
Table of Contents
Learning Objectives
Evaluating Expressions
Solving Equations
Solving Inequalities
Practice
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2
3
4
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Learning Objectives
LO 1 Understand the difference between evaluating and solving
TOC
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
TOC
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
TOC08/09/12
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Step1 Evaluate means you need two thingsAn expression or equationSome value
TOC08/09/12
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
TOC08/09/12
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Step1 Solve means LOLLOL means Letter on LeftEverything else goes to the right by using reverse operations!
TOC08/09/12
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
TOC08/09/12
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Step1 Solve means LOLLOL means Letter on LeftEverything else goes to the right by using reverse operations!
TOC08/09/12
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
TOC08/09/12
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Step1 Inequalities are solved like equationsLOL means Letter on LeftEverything else goes to the right by using reverse operations!
TOC08/09/12
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
TOC08/09/12
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Step1 Inequalities are solved like equationsLOL means Letter on LeftEverything else goes to the right by using reverse operations!
TOC08/09/12
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
TOC08/09/12
lntaylor ©
Step1 Inequalities are solved like equationsLOL means Letter on LeftEverything else goes to the right by using reverse operations!
TOC08/09/12
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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08/09/12 lntaylor ©TOC
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