Difference between revisions of "Rational Equations"

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# The least common denominator of all terms in the equation is <math> x^2 </math>.
 
# The least common denominator of all terms in the equation is <math> x^2 </math>.
 
# Multiplying each side of the equation <math> 1 - \frac{1}{x} = \frac{2}{x^2} </math> with <math> x^2 </math> gives us <math> x^2 - x = 2 </math>
 
# Multiplying each side of the equation <math> 1 - \frac{1}{x} = \frac{2}{x^2} </math> with <math> x^2 </math> gives us <math> x^2 - x = 2 </math>
# <math> x^2 - x = 2 &nbsp;\to &nbsp; x^2 - x - 2 = 0 &nbsp;\to &nbsp; (x - 2)(x + 1) = 0 &nbsp;\to &nbsp; x = -1, x = 2 </math>
+
# <math> x^2 - x = 2 \to x^2 - x - 2 = 0 \to (x - 2)(x + 1) = 0 \to x = -1, x = 2 </math>
 
# None of these solutions were noted in step 1, so we can check our two solutions:
 
# None of these solutions were noted in step 1, so we can check our two solutions:
 
: <math>x = -1</math>:  
 
: <math>x = -1</math>:  

Revision as of 10:55, 22 September 2021

Rational equations are equations containing rational expressions (or expressions with fractions that contain real numbers and/or variables). Some examples of rational equations:

Steps to solving rational equations:

  1. Note any value of the variable that would make any denominator zero.
  2. Find the least common denominator of all denominators in the equation.
  3. Clear the fractions by multiplying both sides of the equation by the LCD.
  4. Solve the resulting equation.
  5. Check: If any values found in step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation.

Example problem:

  1. If x = 0, the denominator of and will be 0.
  2. The least common denominator of all terms in the equation is .
  3. Multiplying each side of the equation with gives us
  4. None of these solutions were noted in step 1, so we can check our two solutions:
:

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