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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
The result of the operation \(\frac{3x}{x-3} - \frac{x+4}{x+2}\) is \(\frac{(2x-3)(x+4)}{(x-3)(x+2)}\).

Step by step solution

01

Identify the denominators

The denominators of the fractions are \(x-3\) and \(x+2\). In order to add or subtract fractions, they must have a common denominator.
02

Find the common denominator

To find a common denominator, the denominators can be multiplied together to get \((x-3) * (x+2)\) as the common denominator.
03

Rewrite the fractions with the common denominator

Rewrite the fractions such that they both have the common denominator. This gives \(\frac{3x * (x+2)}{(x-3)*(x+2)} - \frac{(x+4) * (x-3)}{(x-3)*(x+2)}\).
04

Expand the numerators

Expand the numerators: \(\frac{3x*x+6x}{(x-3)*(x+2)} - \frac{x*x -3x+12}{(x-3)*(x+2)}\) which simlifies to \(\frac{3x^2+6x}{x^2 -x -6} - \frac{x^2 -3x +12}{x^2 -x -6}\).
05

Subtract the fractions

Now the fractions can be subtracted since they have the same denominator. This gives \(\frac{3x^2+6x - (x^2 -3x +12)}{x^2 -x -6}\) which simplifies to \(\frac{2x^2+9x -12}{x^2 -x -6}\).
06

Factorize the numerator and denominator, if possible

In this case, the numerator and denominator can be factorized, giving \(\frac{(2x-3)(x+4)}{(x-3)(x+2)}\).

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Most popular questions from this chapter

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (IMAGE CANT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(\frac{x^{-\frac{5}{4} y^{\frac{1}{3}}}}{x^{-\frac{3}{4}}}\right)^{-6} $$

Describe the kinds of numbers that have rational fifth roots.

Simplify using properties of exponents. $$\frac{\left(3 y^{\frac{1}{4}}\right)^{3}}{y^{\frac{1}{12}}}$$

Simplify by reducing the index of the radical. $$\sqrt[4]{x^{12}}$$
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