Exercise 46 Page 697

We want to solve the given rational equation. x-6/x+3+2x/x-3=4x+3/x+3 Let's begin by highlighting all of the different factors in the denominators. This will help us find the least common denominator (LCD). x-6/x+3+2x/x-3=4x+3/x+3We will solve the equation by multiplying each side by the LCD to clear denominators.

x-6/x+3+2x/x-3=4x+3/x+3

MultEqn

LHS * (x+3) (x-3)=RHS* (x+3) (x-3)

(x-6/x+3+2x/x-3)* (x+3) (x-3)=4x+3/x+3* (x+3) (x-3)

Distr

Distribute (x+3)(x-3)

x-6/x+3* (x+3)(x-3)+2x/x-3* (x+3)(x-3)=4x+3/x+3* (x+3)(x-3)

MoveRightFacToNum

a/c* b = a* b/c

(x-6)(x+3)(x-3)/x+3+2x(x+3)(x-3)/x-3=(4x+3)(x+3)(x-3)/x+3

CancelCommonFac

Cancel out common factors

(x-6)(x+3)(x-3)/x+3+2x(x+3)(x-3)/x-3=(4x+3)(x+3)(x-3)/x+3

SimpQuot

Simplify quotient

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

Distr

Distribute (x-3)

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

▼

Distribute x & -6 & 2x & 4x & 3

Distr

Distribute x

x^2-3x-6(x-3)+2x(x+3)=4x(x-3)+3(x-3)

Distr

Distribute -6

x^2-3x-6x+18+2x(x+3)=4x(x-3)+3(x-3)

Distr

Distribute 2x

x^2-3x-6x+18+2x^2+6x=4x(x-3)+3(x-3)

Distr

Distribute 4x

x^2-3x-6x+18+2x^2+6x=4x^2-12x+3(x-3)

Distr

Distribute 3

x^2-3x-6x+18+2x^2+6x=4x^2-12x+3x-9

▼

Simplify

AddSubTerms

Add and subtract terms

3x^2-3x+18=4x^2-9x-9

SubEqn

LHS-3x^2=RHS-3x^2

-3x+18=x^2-9x-9

AddEqn

LHS+3x=RHS+3x

18=x^2-6x-9

SubEqn

LHS-18=RHS-18

0=x^2-6x-27

RearrangeEqn

Rearrange equation

x^2-6x-27=0

We obtained a quadratic equation. Let's identify the values of a, b, and c. x^2-6x-27=0 ⇕ 1x^2+( - 6)x+( -27)=0 We see that a = 1, b = - 6, and c = -27. Next, we will substitute these values into the Quadratic Formula.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- ( - 6)±sqrt(( - 6)^2-4( 1)( -27))/2( 1)

▼

Solve using the quadratic formula

NegNeg

- (- a)=a

x=6±sqrt((- 6)^2-4(1)(-27))/2(1)

NegBaseToPosBase

(- a)^2=a^2

x=6±sqrt(36-4(1)(-27))/2(1)

IdPropMult

Identity Property of Multiplication

x=6±sqrt(36-4(-27))/2

MultNegNegOnePar

- a(- b)=a* b

x=6±sqrt(36+108)/2

AddTerms

Add terms

x=6±sqrt(144)/2

CalcRoot

Calculate root

x=6± 12/2

We will find the values for x by using the positive and the negative signs.

x=6± 12/2
x=6+ 12/2	x=6- 12/2
x=18/2	x=-6/2
x=9	x=-3

We found that x=9 and x=-3 are possible solutions. Now we have to check them!

Checking the Solutions

We check our solutions to see whether any of them are extraneous. To do so, we will substitute x=9 and x=-3 into the given equation. Let's start with x=9.

x-6/x+3+2x/x-3=4x+3/x+3

Substitute

x= 9

9-6/9+3+2( 9)/9-3? =4( 9)+3/9+3

▼

Simplify

Multiply

9-6/9+3+18/9-3? =36+3/9+3

AddSubTerms

Add and subtract terms

3/12+18/6? =39/12

ExpandFrac