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Chapter 0: Problem 46

Add or subtract as indicated. $$\frac{3 x}{x-3}-\frac{x+4}{x+2}$$

Short Answer

Expert verified
So, \[\frac{3x}{x-3}-\frac{x+4}{x+2} = \frac{2x^2+7x+12}{(x-3)(x+2)}\].

Step by step solution

01

Identify Common Denominator

The common denominator between a variable subtracted from a constant (\(x-3\)) and variable added to a constant (\(x+2\)) is their product (\((x-3)(x+2)\)). So let's multiply the fractions to have this common denominator.
02

Multiply the Fractions by the Corresponding Denominator

Multiply \(\frac{3x}{x-3}\) by \(\frac{x+2}{x+2}\) and \(\frac{x+4}{x+2}\) by \(\frac{x-3}{x-3}\) to have the common denominator. We then get: \[\frac{3x(x+2)}{(x-3)(x+2)}-\frac{(x+4)(x-3)}{(x-3)(x+2)}\].
03

Subtract the Numerators

Now that the fractional expressions have a common denominator, continue the subtraction by subtracting the numerators: \[\frac{3x^2+6x-(x^2-x-12)}{(x-3)(x+2)}\] or \[\frac{3x^2+6x-x^2+x+12}{(x-3)(x+2)}\].
04

Simplify the Numerator

Simplify the numerator by combining similar terms: \[\frac{2x^2+7x+12}{(x-3)(x+2)}\].

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Most popular questions from this chapter

What is a rational expression?

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\).

Explain how to simplify a rational expression.
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