Solving rational equations

Solving Rational Equations

Rational Equations with more than 2 terms

• The LCD (lowest common denominator) is the combination of all the denominators of the equations. Reduce this if possible.

• Multiply each term/fraction by the LCD

• Cancel any common factors or terms

• Simplify and solve for x.​

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​When we can't cross multiply, use LCD

Solving rational equations

• 1. Factor the DENOMINATORS only

• 2. Multiply each side of the equation by the least common multiple (LCM) this should get rid of the fraction

• 3. Solve

• 4. Check to make sure your answers work (you don't divide by zero)

Check your answers 

Check

Check:

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Multiple Choice

Solve the following: $$\frac{x}{x^2-49}=\frac{2x}{x+7}$$  

• $$x=0$$  

• $$x=\frac{15}{2}$$  

• $$x=0,\ \frac{15}{2}$$  

• $$extraneous\ \ \ \ solution\left(s\right)$$  

Multiple Select

Solve

$$\frac{x+1}{x-3}=-\frac{x^2-6x+1}{\left(x-3\right)\left(x-1\right)}$$  

• -5

• -2

• 0

• 1

• 4

Multiple Choice

$$\frac{1}{12p^2}=\frac{1}{6p^2}-\frac{1}{p}$$  

• $$-\frac{1}{12}$$  

• 6

• $$\frac{1}{12}$$  

• -2

Multiple Choice

Solve the following: $$\frac{5}{x-2}=\frac{1}{x+3}$$  

• $$x=\frac{17}{4}$$  

• $$x=\frac{-17}{4}$$  

• $$x=\frac{4}{17}$$  

• $$extraneous\ \ \ \ solution\left(s\right)$$  

Multiple Choice

Solve the following: $$\frac{x}{x+3}=\frac{8}{x+6}$$  

• $$x=-4$$  

• $$x=6$$  

• $$x=-4,\ 6$$  

• $$extraneous\ \ \ \ solution\left(s\right)$$  

Multiple Choice

Solve the following: $$\frac{x}{x-2}+\frac{1}{5}=\frac{2}{x-5}$$  

• $$x=2$$  

• $$x=\frac{1}{2}$$  

• $$x=-2$$  

• $$extraneous\ \ \ \ solution\left(s\right)$$  

Multiple Choice

Which of the following examples of Rational Equations can be solved by cross multiplication?

• $$\frac{x}{x-2}=\frac{3}{5}$$

• $$\frac{x}{x+1}\ -3x\ =\ \frac{2}{x-1}$$

• $$\frac{1}{x}+\frac{3}{2x\ }=\ \frac{1}{3x}$$

• $$x\ +\frac{2}{2x}=\frac{2}{x-3}$$

Multiple Choice

Which of the following equation is correct after cross multiplication:

$$\frac{2}{x-3}=\frac{3}{x-2\ ?}$$  

• $$3x\ =\ 2x\ -4$$  

• $$3x\ -9\ =\ 2x\ -\ 4$$  

• $$3x\ +4\ =\ 2x\ -9$$  

• $$3x\ -3\ =\ x\ -2$$