Rational Equation (no solution)

Example: Solve the rational equation.




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.




Having x=3 causes a zero in the denominator and the overall expression undefined.  That makes 3 a restricted value .

With the restriction in mind we will solve the equation.


The original equation
Multiply each side of the equation by the least common multiple of the denominators.  For this equation the least common multiple is x-3
Distribute the least common multiple to each term.
Simplify by canceling the common factors.  This should clear any denominators.
Use the distributive property to simplify.
Simplify each side of the equation by combining like terms.
Solve for x by getting x by itself on one side.  Start by subtracting 1 on both sides.
 Solve for x by getting x by itself on one side.  Next divide both sides by 2.
Compare your solution to the restricted value.
Since the solution is the same as the restricted value we must exclude it as a solution.  Since all of the solutions have been excluded, there is no solution to the rational equation.

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