Rational Equations with Extraneous Solutions

6.4 Part III - Solve Rational Equations

Rational Equation

A rational equation is an equation that contains one or more rational expressions.

Let's think back to when we learned how to add fractions

• We need a common denominator, right?

•  $\frac{1}{2}+\frac{1}{3}=?$  

• What is the LCM of 2 & 3? Six, right? So our common denominator is 6.

• So we multiply the top and bottom of both fractions by the "missing factor". $\frac{1\cdot3}{2\cdot3}+\frac{1\cdot2}{3\cdot2}=\frac{3+2}{6}=\frac{5}{6}$  

Example

•  $\frac{6}{x}+\frac{5}{4}=-\frac{7}{4}$  

• We have an x and a 4 for denominators, so our LCM is 4*x = 4x

•  $\frac{6\cdot4}{x\cdot4}+\frac{5\cdot x}{4\cdot x}=-\frac{7\cdot x}{4\cdot x}$  

•  $\frac{24}{4x}+\frac{5x}{4x}=-\frac{7x}{4x}$  

•  $24+5x=-7x$  

•  $24=-12x$  

•  $-2=x$  

Recall that we have to test for extraneous solutions.

Multiple Choice

Solve the rational equation $$\frac{16}{x^2-16}=\frac{2}{x-4}$$  

• $$x=-4;\ x=4\ is\ an\ extraneous\ solution$$  

• There are no solutions because  $$x=4$$  is an extraneous solution

• There are no solutions because  $$x=16$$  is an extraneous solution

• $$x=4,\ x=\ -4$$  

You try

Open Ended

Solve the rational equation $$4=\frac{3}{2-c}+\frac{3}{2+c}$$  

Some quick practice on MathXL and you're good to go for the day.