Q:

# How do you solve rational equations?

A:

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.

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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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