How Do You Solve Rational Equations?

# How Do You Solve Rational Equations?

To solve rational equations, use the common denominator to resolve all the fractions. Removing the denominators makes it much easier to deal with the remaining terms, converting the problem into a simpler equation.

1. Find the common denominator

Consider a very simple example: 9/10 = 3x/10. Realize that the common denominator is 10 and multiply both sides by 10 to yield 9 = 3x. Divide both sides by 3 for the solution: x = 3. Consider a slightly more complex example: (x-4)/12 = 1/4. Identify 4 as a factor of 12, making 12 the common denominator. Consider an even more complex example: 4/(x+3) - 2/x = 3/4x. Multiply (x+3) by 4x to get 4x(x+3) which is the common denominator, since x is a factor of 4x.

2. Multiply through by the common denominator

Take the second example from Step 1. Multiply the right side of the equation by 3/3 to elevate the denominator to 12 without altering the balance of the equation. Remember that 3/3 = 1 and that multiplying by 1 does not change the value of a term. Write the next step as (x-4)/12 = 3/12. Take the third example from Step 1. Multiply every term by [4x(x+3)]/1. Since you are multiplying every term in the equation by the same value, the equation remains unchanged in relative value. Write the next step as [4/(x+3)][4x(x+3)/1] - [2/x][4x(x+3)/1] = [3/4x][4x(x+3)/1].

3. Simplify and solve the equations

Conclude the first example from Step 2. Multiply both sides by 12 to yield x-4 = 3. Add 4 to both sides for the solution: x = 7. Conclude the second example from Step 2. Cross out like numerators and denominators in each term to yield 16x - 8(x+3) = 3(x+3). Expand the parentheses to yield 16x - 8x - 24 = 3x + 9. Combine like terms to yield 8x - 24 = 3x + 9. Subtract 3x from both sides to yield 5x - 24 = 9. Add 24 to both sides to yield 5x = 15, and divide both sides by 5 for the solution: x = 3.

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