Solving More Complex Rational Equations

Description/Explanation/Highlights

Video Description

This video will explain how to solve more complex rational equations that require factoring and simplifying.

Steps and Key Points to Remember

To solve rational equations that require simplifying denominators first, follow these steps:

  1. Remember, a rational equation is any equation that has a variable in the denominator.
  2. \(\frac{\textstyle 3}{\textstyle x^2+5x+6}+\frac{\textstyle x-1}{\textstyle x+2}=\frac{\textstyle 7}{\textstyle x+3}\) is an example of a rational equation. There are multiple \(x\)s in the denominators of the expressions on both sides of the equal sign.
  3. Start by simplifying any expression that can be simplified. Since the denominator of the first expression in the example above can be factored, it should be simplified by factoring first. The result after factoring will be: \(\frac{\textstyle 3}{\textstyle (x+3)(x+2)}+\frac{\textstyle x-1}{\textstyle x+2}=\frac{\textstyle 7}{\textstyle x+3}\)
  4. As with any rational equation, after simplifying, 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.
  5. To find these restrictions, set each unique factor in the denominator equal to zero and solve for the variable. In the example above set \(x+3=0\) and \(x+2=0\).
  6. Solve each for \(x\). In the example \(x=-3\) and \(x=-2\) therefore the solution of x in the rational equation cannot be -3 or -2 and if solving the equation results in one of these answers, that answer must be eliminated.
  7. Find the least common denominator (LCD) by listing each unique factor in the denominators. In the above example, \(x+3\) and \(x+2\) in the first expression are unique so \((x+3)(x+2)\) is the LCD.
  8. Multiply both sides of the equation by the LCD. The effect of this will be to eliminate the denominators since the factors will cancel.
  9. In the above example \((x+3)(x+2)(\frac{\textstyle 3}{\textstyle (x+3)(x+2)}+\frac{\textstyle x-1}{\textstyle x+2})=(\frac{\textstyle 7}{\textstyle x+3})(x+3)(x+2)\) after canceling will become: \(3+(x-1)(x+3)=7(x+2)\).
  10. Solve the resulting equation as you would any equation by distributing and solving. In this example, distribute the 7 on the right side and multiply the binomials together on the left to get \(3+x^2+2x-3=7x+14\).
  11. The \(x^2\) tells us this is a quadratic so combine terms to one side and set equal to zero and solve. \(x^2-5x-14=0\)
  12. To solve a quadratic, factor if possible and set each factor equal to zero and solve for the values of x. If the quadratic cannot be factored, use the quadratic formula to solve for the x-values. 
  13. The above example will factor as \((x-7)(x+2)=0\).
  14. Set \(x-7=0\) and \(x+2=0\). The resulting x-values will be \(x=7\) and \(x=-2\).
  15. Check domain restrictions that were calculated earlier. The answer we found, 7, is not one of the restricted domain of -2 or -3 so it is a valid solution but -2 is an answer that is not allowed and must be eliminated. The only valid solution is \(x=7\).

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

  • Before solving, simplify the expressions by factoring if possible.
  • Always start the solution to a rational equation by finding out solutions that will not be valid by setting each factor in the 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 answers are not one of the restrictions and are valid answers.

Video Highlights

  • 00:00 Introduction
  • 00:10 \(\frac{\textstyle 3}{\textstyle x^2+5x+6}+\frac{\textstyle x-1}{\textstyle x+2}=\frac{\textstyle 7}{\textstyle x+3}\) example of solving a rational equation that requires simplification.
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