Rational Equations and Problem Solving
A Rational equation is an equation that contain at least one rational expression. In order to solve a rational expression, one has to do the following steps:
Step 1) Multiply all the terms by the least common denominator to eliminate the denominators.
Step 2) Simplify the equation obtained in step 1.
Step 3) Solve the simplified equation for the variable.
Step 4) Check the solution to make sure that it satisfies the original equation.
Example: Solve
.
Step 1) The least common denominator for the denominators in this example is
. Therefore, we multiply all the terms by:
Step 2) We simplify the equation from step 1:
Step 3) We solve for x:
Step 4) We verify the solution by substituting the solution into the original equation
Since the left-hand-side of the equation is equal to the right-hand-side of the equation, the solution is valid.
Example: Solve
Step 1) We have to multiply everything by the least common denominator. To do so, we have to factor denominators and find the least common denominator for them:
now, we substitute the factored form of the denominators into the original equation:
The least common denominator is
. So, we multiply everything by the least common denominator:
Step 2) We simplify the above equation:
Step 3) We solve the equation from step 2:
Step 4) We verify the solution:
after simplifying, we get:
Since the left-hand-side is equal to the right-hand-side, the solution is acceptable.
