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Chapter 0: Problem 53

Add or subtract as indicated. $$\frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6}$$

Short Answer

Expert verified
The simplified form of the given expression \(\frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6}\) is \(\frac{x(x-1)}{(x-2)(x+5)(x+3)}\).

Step by step solution

01

Factor the Denominators

Factor each denominator so that they can be expressed in terms of their factors. For \(x^{2}+3 x-10\), its factors are \(x-2\) and \(x+5\). For \(x^{2}+x-6\), its factors are \(x-2\) and \(x+3\). So the expressions become: \[\frac{3x}{(x-2)(x+5)} - \frac{2x}{(x-2)(x+3)}.\]
02

Find the Least Common Denominator (LCD)

The LCD of 2 fractions is the smallest expression that is a multiple of each denominator. Here LCD is the product of all un-common factors, which are \((x-2)\), \((x+5)\), and \((x+3)\). So, LCD is \((x-2)(x+5)(x+3)\).
03

Rewrite the Fractions

Rewrite the fractions such that they have the LCD as the denominator. Multiply numerator and denominator of first fraction by \((x+3)\), and multiply numerator and denominator of second fraction by \((x+5)\): \[\frac{3x*(x+3)}{(x-2)(x+5)(x+3)} - \frac{2x*(x+5)}{(x-2)(x+5)(x+3)}.\]
04

Simplify the Expressions

Now simplify the expressions in the numerators and combine these two fractions as they have same denominator now: \[\frac{3x^{2} +9x -2x^{2} -10x}{(x-2)(x+5)(x+3)} = \frac{x^{2} -x}{(x-2)(x+5)(x+3)}.\]
05

Factor and Simplify Final Expression

Factor the term in numerator and simplify the final result as: \[\frac{x(x -1)}{(x-2)(x+5)(x+3)},\] This is the final simplified version of the expression.

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Most popular questions from this chapter

What is a linear equation in one variable? Give an example of this type of equation.

$$\text { Solve for } C: \quad V=C-\frac{C-S}{L} N$$.

Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\).

Will help you prepare for the material covered in the next section. If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200\), write a simplified algebraic expression that models the rectangle's perimeter.

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$
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