Solve Rational Equations: example 8.64

Solve: $\frac{m + 11}{m^{2} - 5 m + 4} = \frac{5}{m - 4} - \frac{3}{m - 1} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve: $\frac{x + 13}{x^{2} - 7 x + 10} = \frac{6}{x - 5} - \frac{4}{x - 2} .$

How did it go?


Factor all the denominators, so we can note any value of the variable the would make any denominator zero.
Find the least common denominator of all denominators in the equation. The LCD is $(m - 4) (m - 1)$ .
Clear the fractions.
Distribute.
Remove common factors.
Simplify.
Solve the resulting equation.

Check. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.


Factor all the denominators, so we can note any value of the variable the would make any denominator zero.
Find the least common denominator of all denominators in the equation. The LCD is $(m - 4) (m - 1)$ .
Clear the fractions.
Distribute.
Remove common factors.
Simplify.
Solve the resulting equation.

Check. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.