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Core Connections Integrated II, 2015 View details

2. Section 1.2

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Exercise 26 Page 20

Practice makes perfect

a To solve an equation, we should first gather all of the variable terms on one side of the equation and all constants on the other side by using the Properties of Equality. In this case, we will start by multiplying both sides of the equation by the lowest common denominator to remove the fractions.

3x-1/4=- 5/11

▼

LHS * 44=RHS* 44

MultEqn

LHS * 44=RHS* 44

3x-1/4(44)=- 5/11(44)

SplitIntoFactors

Split into factors

3x-1/4(4)(11)=- 5/11(11)(4)

FracMultDenomToNumber

a/4* 4 = a

(3x-1)(11)=- 5/11(11)(4)

FracMultDenomToNumber

a/11* 11 = a

(3x-1)(11)=- 5(4)

Distr

Distribute 11

33x-11=- 5(4)

MultNegPos

(- a)b = - ab

33x-11=- 20

AddEqn

LHS+11=RHS+11

33x=- 9

DivEqn

.LHS /33.=.RHS /33.

x=- 9/33

▼

Simplify

MoveNegNumToFrac

Put minus sign in front of fraction

x=- 9/33

ReduceFrac

a/b=.a /3./.b /3.

x=- 3/11

The solution to the equation is x=- 311. We can check our solution by substituting it into the original equation.

3x-1/4=- 5/11

Substitute

x= - 3/11

3( - 311 ) -1/4 ? = - 5/11

▼

Simplify

MultPosNeg

a(- b)=- a * b

- 3( 311 ) -1/4 ? = - 5/11

MoveLeftFacToNum

a*b/c= a* b/c

- 911 -1/4 ? = - 5/11

NumberToFrac

a = 11* a/11

- 911 - 1111/4 ? = - 5/11

SubFrac

Subtract fractions

- 2011/4 ? = - 5/11

DivFracD

a/c/b= a/b* c

- 20/4(11) ? = - 5/11

ReduceFrac

a/b=.a /4./.b /4.

- 5/11 ? = - 5/11

MoveNegNumToFrac

Put minus sign in front of fraction

- 5/11=- 5/11 ✓

Since the left-hand side is equal to the right-hand side, our solution is correct.

b To solve an equation, we should first gather all of the variable terms on one side of the equation and all of the constant terms on the other side by using the Properties of Equality. In this case, we need to start by using the Distributive Property to simplify the left-hand side of the equation.

6-5(2x-3)=4x+7

Distr

Distribute - 5

6-5(2x)-5(- 3)=4x+7

MultNegPos

(- a)b = - ab

6-10x-5(- 3)=4x+7

MultNegNegOnePar

- a(- b)=a* b

6-10x+15=4x+7

Now we can continue to solve using the Properties of Equality.

6-10x+15=4x+7

AddTerms

Add terms

21-10x=4x+7

SubEqn

LHS-7=RHS-7

14-10x=4x

AddEqn

LHS+10x=RHS+10x

14=14x

RearrangeEqn

Rearrange equation

14x=14

DivEqn

.LHS /14.=.RHS /14.

x=1

The solution to the equation is x=1. We can check our solution by substituting it into the original equation.

6-5(2x-3)=4x+7

Substitute

x= 1

6-5(2( 1)-3) ? = 4( 1)+7

▼

Simplify

IdPropMult

Identity Property of Multiplication

6-5(2-3) ? = 4+7

AddSubTerms

Add and subtract terms

6-5(- 1) ? = 11

MultNegNegOnePar

- a(- b)=a* b

6+5 ? = 11

AddTerms

Add terms

11 = 11 ✓

Since the left-hand side is equal to the right-hand side, our solution is correct.

c To solve an equation, we should first gather all of the variable terms on one side of the equation and all constants on the other side by using the Properties of Equality. In this case, we will start by multiplying both sides of the equation by the lowest common denominator to remove the fractions.

3x/4+2=4x-1

MultEqn

LHS * 4=RHS* 4

3x/4(4)+2(4)=4(4)x-1(4)

Multiply

3x/4(4)+8=16x-4

FracMultDenomToNumber

a/4* 4 = a

3x+8=16x-4

Now we can continue to solve using the Properties of Equality.

3x+8=16x-4

AddEqn