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Solving Rational Equations
Solving rational equations follows the same rules as solving any other type of equation. Whatever you do to one side of the equation, you must do to the other. If you have fractions, you try to eliminate them by multiplying by the common denominator. If there are quadratics involved, you must get all terms to one side with zero on the other. If you need practice solving linear equations (link to linear equations) or solving quadratic equations, click on the link to review those skills before working with rational equations.

Recall that a rational expression is in the form of a fraction where there is a variable in the denominator. Solving rational equations will involve simplifying rational expressions. If you need to review that topic, click here.

Before we begin with a rational function, let’s look at how we would handle an equation like

Our first step in solving this equation would be to multiply each term by the least common denominator of the fractions, which is 12.

Simplifying this particular equation results in the following linear equation which we can then solve to get our final answer.


It was important to look at an equation that did not have a variable in the denominator to make sure we see the pattern for solving rational equations. Here are the steps we will use in our solution process.
  1. Determine the least common denominator of all the fractions in the equation.
  2. Eliminate the fraction(s) by multiplying ALL terms by the least common denominator.
  3. Simplify the terms.
  4. Solve the resulting equation.
  5. Check your answers to make sure the solution does not make the fraction undefined.

Examples
Example
#1 Solve

Step 1: Find the least common denominator (LCD).
LCD =
Step 2: Multiply all terms by the LCD.
Step 3: Simplify and solve for x.
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Example
#2: Solve

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Example
#3: Solve

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#4: Solve

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#5: Solve

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#6: Solve

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#7: Solve

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#8: Solve

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S Taylor

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