Objectives 

• Linear Equations

• Equations with variables in a denominator

Examples

• Example 1

• Example 2

• Example 3

• Example 4

Find all real solutions (if any) for the given equation:



Solution:

Our goal is to "isolate the x."

    Step 1: Use the LCD to clear out the denominators.

        The LCD for the rational expressions in the original equation
         is  .

        Multiplying each side of the equation by the LCD yields



        Now we distribute the LCD through the left hand side of the
        equation.



        In the first term on the left, the (x + 2) cancels and in the second
        term on the left, the (4x - 1) cancels.



    Step 2: Solve the resulting equation.



       Multiply out each side and combine like terms.





        Isolate the terms which contain the variable x.











Our work demonstrates that the equation from Step 2 will yield a 
true statement provided the variable "x" is assigned the number 
value 10/13. 

But be careful: Since the original equation contains denominators involving the variable x,we must check to make sure our answer from Step 2 does not cause any of the denominators to equal zero.

In this example, the only values of x which yield "denominator equal zero" are x = - 2 and x = 1/4.

Therefore,  is a true solution to the original equation.