1.1 Linear Equations

Rational Equation

Example: Solve the rational equation.

{2x}/{x^2-7x-8}-7/{x-8}=1/{x+1}

Solution:

{2x}/{x^2-7x-8}-7/{x-8}=1/{x+1}

Since we are solving a rational equation we need to first find the restrictions (the values of x that cause the expression to be undefined).

To find the restrictions create an equation by setting each denominator equal to zero and solving.

x-8=0

x-8+8=0+8

x=8

Having x=8 causes a zero in the denominator and the overall expression undefined.  That makes 8 a restricted value .

x+1=0

x+1-1=0-1

x=-1

Having x=-1 causes a zero in the denominator and the overall expression undefined.  That makes -1 a restricted value .

x^2-7x-8=0

(x-8)(x+1)=0

x-8=0 or x+1=0

x=8 or x=-1

This gives the same restrictions we have already accounted for.

With the restriction in mind we will solve the equation.

 

The original equation
 {2x}/{x^2-7x-8}-7/{x-8}=1/{x+1}
The original equation with each denominator factore
 {2x}/{(x-8)(x+1)}-7/{x-8}=1/{x+1}
Multiply each side of the equation by the least common multiple of the denominators.  For this equation the least common multiple is (x-8)(x+1)
{2x}/{(x-8)(x+1)}-7/{x-8}=1/{x+1}
(x-8)(x+1){2x}/{(x-8)(x+1)}-(x-8)(x+1) 7/{x-8}=(x-8)(x+1) 1/{x+1}
Simplify by canceling the common factors.  This should clear any denominators.
(x-8)(x+1){2x}/{(x-8)(x+1)}-(x-8)(x+1) 7/{x-8}=(x-8)(x+1) 1/{x+1}
2x-7(x+1) =x-8
Use the distributive property to simplify.
2x-7(x+1) =x-8
2x-7x-7 =x-8
Simplify each side of the equation by combining like terms.
2x-7x-7 =x-8
-5x-7 =x-8
Solve for x by getting the variables on one side.
-5x-7 =x-8
-5x-x-7 =x-x-8
-6x-7 =-8
Solve for x by getting x by itself on one side.  Start by adding 7 on both sides.
-6x-7 =-8
-6x-7+7 =-8+7
-6x =-1
 Solve for x by getting x by itself on one side.  Next divide both sides by -6.
-6x =-1
{-6x}/-6=-1/-6
x=1/6
Compare your solution to the restricted value.
Since the solution is not the same as the restricted value we can consider it a solution to the original equation

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