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McGraw Hill Glencoe Algebra 1, 2012 View details

8. Rational Equations

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Exercise 6 Page 730

Factor the denominators and multiply both sides of the equation by the least common denominator.

x=-4/3

Practice makes perfect

We will solve the given equation and then check the solutions.

Solving the Equation

We want to solve the given rational equation. x+3/x^2-1-2x/x-1=1 We will start by factoring the denominators to find the least common denominator (LCD). Note that the second denominator is already factored. Let's factor the first one.

x^2-1

WritePow

Write as a power

x^2-1^2

FacDiffSquares

a^2-b^2=(a+b)(a-b)

(x+1)(x-1)

We can now rewrite the given expression.

x+3/x^2-1-2x/x-1=1 ⇕ x+3/( x+1)( x-1)-2x/x-1=1 We can see that the LCD is equal to ( x+1)( x-1). We will solve the equation by multiplying each side by the LCD to clear the denominators.

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

MultEqn

LHS * ( x+1)( x-1)=RHS* ( x+1)( x-1)

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

Multiply

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

CancelCommonFac

Cancel out common factors

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

SimpQuot

Simplify quotient

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

▼

Simplify

Distr

Distribute -2x

x+3-2x^2-2x=(x+1)(x-1)

ExpandDiffSquares

(a+b)(a-b)=a^2-b^2

x+3-2x^2-2x=x^2-1^2

CalcPow

Calculate power

x+3-2x^2-2x=x^2-1

AddEqn

LHS+1=RHS+1

x+3-2x^2-2x+1=x^2

SubEqn

LHS-x^2=RHS-x^2

x+3-2x^2-2x+1-x^2=0

CommutativePropAdd

Commutative Property of Addition

-2x^2-x^2-2x+x+3+1=0

AddSubTerms

Add and subtract terms

-3x^2-x+4=0

Note that we have a quadratic equation now. Let's identify the values of a, b, and c. -3x^2-x+4=0 ⇔ -3x^2+( -1)x+ 4=0 We have that a= - 3, b= -1, and c= 4. Let's substitute these values into the Quadratic Formula and solve for x.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- ( -1)±sqrt(( -1)^2-4( - 3)( 4))/2( - 3)

▼

Simplify right-hand side

CalcPowProd

Calculate power and product

x=-(-1)±sqrt(1+48)/- 6

NegNeg

- (- a)=a

x=1±sqrt(1+48)/- 6

AddTerms

Add terms

x=1±sqrt(49)/- 6

CalcRoot

Calculate root

x=1± 7/- 6

Let's calculate both solutions by using the positive and negative signs.

x=1± 7/- 6
x=1+ 7/- 6	x=1- 7/- 6
x=8/- 6	x=- 6/- 6
x=-4/3	x=1

We found that x= -43 and x=1 are possible solutions to the equation. Now we have to check them!

Checking the Solutions

We will check if either x=- 43 or x=1 is an extraneous solution. To do this, we need to substitute - 43 and 1 for x in the original equation. Let's start with x=1.

x+3/x^2-1-2x/x-1=1

Substitute

x= 1

1+3/1^2-1-2( 1)/1-1? =1

CalcPowProd

Calculate power and product

1+3/1-1-2/1-1? =1

SubTerms

Subtract terms

1+3/0-2/0≠ 1 *

Since x=1 results in a zero in denominator of the original equation, it is an extraneous solution. Now we will check our second solution, x=- 43.

x+3/x^2-1-2x/x-1=1

Substitute

x= -4/3

- 43+3/( - 43)^2-1-2( - 43)/- 43-1? =1

CalcPowProd

Calculate power and product

- 43+3/169-1-- 83/- 43-1? =1

NumberToFrac

a = 3* a/3

- 43+ 93/169-1-- 83/- 43- 33? =1

NumberToFrac

a = 9* a/9

- 43+ 93/169- 99-- 83/- 43- 33? =1

AddSubFrac

Add and subtract fractions

53/79-- 83/- 73? =1

Recall that dividing by a fraction is the same as multiplying by its reciprocal. 53/79-- 83/- 73? =1 ⇕ 5/3(9/7)-(-8/3)(-3/7)? =1 Let's simplify to finally see whether the equation is satisfied when x=- 43.

5/3(9/7)-(-8/3)(-3/7)? =1

▼

Simplify

MultNegNegTwoPar

(- a)(- b)=a* b

5/3(9/7)-8/3*3/7? =1

MultFrac

Multiply fractions

45/21-24/21? =1

SubFrac

Subtract fractions

21/21? =1