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Core Connections Geometry, 2013
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Core Connections Geometry, 2013 View details
3. Section 1.3
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Exercise 114 Page 65

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 this equation, we should apply the Zero Product Property. After obtaining two equations, in both of them we will 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.
(5-x)(2x+3)=0
ZeroProdProp

Use the Zero Product Property

l5-x=0 2x+3=0
Now we can continue to solve using the Properties of Equality.
lc5-x=0 & (I) 2x+3=0 & (II)
▼
(I): Solve for x
AddEqn

(I): LHS+x=RHS+x

l5=x 2x+3=0
RearrangeEqn

(I): Rearrange equation

lx=5 2x+3=0
▼
(II): Solve for x
SubEqn

(II): LHS-3=RHS-3

lx=5 2x=- 3
DivEqn

(II): .LHS /2.=.RHS /2.

lx=5 x= - 32
MoveNegNumToFrac

Put minus sign in front of fraction

lx_1=5 x_2=- 32
The solutions to the equation are 5 and - 32. We can check our solution by substituting them into the original equation. Let's start with x=5.
(5-x)(2x+3)=0
Substitute

x= 5

(5- 5)(2( 5)+3) ? = 0
▼
Simplify
SubTerm

Subtract term

(0)(2(5)+3) ? = 0
ZeroPropMult

Zero Property of Multiplication

0=0 ✓
Since substituting and solving resulted in a true statement, we know that x=5 is a solution of the equation. Let's now check x=- 32.
(5-x)(2x+3)=0
Substitute

x= - 3/2

(5-( - 3/2))(2( - 3/2)+3) ? = 0
▼
Simplify
MultPosNeg

a(- b)=- a * b

(5-(- 3/2))(- 2 ( 3/2)+3) ? = 0
DenomMultFracToNumber

2 * a/2= a

(5-(- 3/2))(- 3+3) ? = 0
AddTerms

Add terms

(5-(- 3/2))(0) ? = 0
ZeroPropMult

Zero Property of Multiplication

0=0 ✓
This is also a true statement, so we know that x=- 32 is a solution of the equation.
c 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.
d 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

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

LHS+4=RHS+4

3x+12=16x
SubEqn

LHS-3x=RHS-3x

12=13x
DivEqn

.LHS /13.=.RHS /13.

12/13=x
RearrangeEqn

Rearrange equation

x=12/13
The solution to the equation is x= 1213. We can check our solution by substituting it into the original equation.
3x/4+2=4x-1
Substitute

x= 12/13

3( 1213 )/4+2 ? = 4( 12/13 )-1
▼
Simplify
MoveLeftFacToNum

a*b/c= a* b/c

3613/4+2 ? = 48/13-1
DivFracD

a/c/b= a/b* c

36/4(13)+2 ? = 48/13-1
ReduceFrac

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

9/13+2 ? = 48/13-1
NumberToFrac

a = 13* a/13

9/13+26/13 ? = 48/13-13/13
AddSubFrac

Add and subtract fractions

35/13=35/13 ✓
Since the left-hand side is equal to the right-hand side, our solution is correct.
Subchapter links expand_more
arrow_right 1.3.1 Attributes and Characteristics of Shapes p.64-65
112
Attributes and Characteristics of Shapes 113 (Page 65)
Attributes and Characteristics of Shapes 114 (Page 65)
115
Attributes and Characteristics of Shapes 116 (Page 65)
arrow_right 1.3.2 More Characteristics of Shapes p.68-69
123
124
More Characteristics of Shapes 125 (Page 69)
126
127