Solving Rational Equations

Description/Explanation/Highlights

Video Description

This video will explain how to solve a rational equation.

Steps and Key Points to Remember

To solve rational equations, follow these steps:

  1. A rational equation is any equation that has a variable in the denominator.
  2. \(\frac{\textstyle 5}{\textstyle x-2}=\frac{\textstyle 6}{\textstyle x+1}\) is an example of a rational equation. There is an \(x\) in the denominator of the expressions on both sides of the equal sign.
  3. Before beginning to solve a rational equation, we must first determine what domain restrictions exist. These are values of x that cannot be used as a solution because they would create an undefined value; in this case a zero in the denominator.
  4. To find these restrictions, set each denominator equal to zero and solve for the variable. In the example above set \(x-2=0\) and \(x+1=0\).
  5. Solve each for \(x\). In the example \(x=2\) and \(x=-1\) therefore the solution of x in the rational equation cannot be 2 or -1 and if solving the equation results in one of these answers, that answer must be eliminated.
  6. Find the least common denominator (LCD) by listing each unique factor in the denominators. In the above example, \(x-2\) in the first expression and \(x+1\) in the second expression are both unique so \((x-2)(x+1)\) is the LCD.
  7. Multiply both sides of the equation by the LCD. The effect of this will be to eliminate the denominators since the factors will cancel.
  8. In the above example \((x-2)(x+1)(\frac{\textstyle 5}{\textstyle x-2})=(\frac{\textstyle 6}{\textstyle x+1})(x-2)(x+1)\)
  9. The \(x-2\) factor will cancel with the denominator on the left side and \(x+1\) on the right side, leaving \(5(x+1)=6(x-2)\).
  10. Solve the resulting equation as you would any linear equation by distributing and solving. In this example, distribute the 5 on the left side and 6 on the right side to get \(5x+5=6x-12\).
  11. Get \(x\) on one side and the numbers on the other by subtracting 5x from both sides and adding 12 to both sides to get the final answer of \(17=x\) or \(x=17\). 
  12. Check domain restrictions that were calculated earlier. the answer we found, 17, is not one of the restricted domain of 2 or -1 so it is a valid solution.
  13. If there is more than just one term on a side such as \(\frac{\textstyle 12}{\textstyle x-4}+\frac{\textstyle 3}{\textstyle x}=\frac{\textstyle 9}{\textstyle x}\), find the domain restrictions for all the denominators by setting each unique denominator = 0 and solving for x as before. In this example x cannot equal 0 or 4.
  14. Find the LCD for all denominators by listing all the unique factors in all the denominators.
  15. In this example \(x-4\) and \(x\) are unique from the first two denominators but \(x\) has already been used so the third denominator has no unique factors. The LCD is \(x(x-4)\).
  16. After multiplying each term by the LCD and canceling the denominators, we are left with \(12x+3(x-4)=9(x-4)\). Notice how the numerators are always multiplied by the parts of the LCD that were not in the original denominator and the original denominator is always canceled. This is a shortcut that can be used to our advantage.
  17. Distribute, solve, and check the answer against the domain restrictions as before and the final answer is \(x=-4\). Due to domain restrictions, the answer could not be 0 or 4 but since it is -4, the answer is valid.

Here are some key points to keep in mind when solving rational equations.

  • Always start the solution to a rational equation by finding out solutions that will not be valid by setting each denominator equal to 0 and solving.
  • These solutions are restrictions to the domain and are not valid solutions to the equation.
  • Find the LCD (least common denominator) by listing all unique factors from all denominators in the equation.
  • Multiple each term in the equation by the LCD.
  • Denominators will be eliminated and numerators will be multiplied by the part of the LCD that was not in the original denominator.
  • It may be a helpful shortcut to drop the denominators and multiply the numerator of each term by any part of the LCD missing in the original denominator for that term.
  • Find the answer by distributing and solving as you would any equation.
  • Check the answer against the domain restrictions you found to be sure your answer is not one of the restrictions and is a valid answer.

Video Highlights

  • 00:00 Introduction
  • 00:12 \(\frac{\textstyle 5}{\textstyle x-2}=\frac{\textstyle 6}{\textstyle x+1}\) example of solving a rational equation.
  • 05:10 \(\frac{\textstyle 12}{\textstyle x-4}+\frac{\textstyle 3}{\textstyle x}=\frac{\textstyle 9}{\textstyle x}\) example of solving a rational equation with more than one term on a side of the equation.
  • 09:10 Conclusion
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