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Chapter 0: Problem 41

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{1}{x-3}+\frac{1}{x+3}=\frac{10}{x^{2}-9} $$

Short Answer

Expert verified
The solution to the equation is \(x = 5\).

Step by step solution

01

Simplify the equation

In the equation \( \frac{1}{x-3} + \frac{1}{x+3} = \frac{10}{x^2 - 9} \), start by simplifying the left side by finding a common denominator. Note that the common denominator on the right is \(x^{2}-9\) which is equivalent to \((x-3)(x+3)\). Multiply the first fraction by \(\frac{x+3}{x+3}\) and the second fraction by \(\frac{x-3}{x-3}\). You get \(\frac{x+3}{(x-3)(x+3)} + \frac{x-3}{(x-3)(x+3)}\).
02

Combine terms and solve

Combine the fractions on the left side to get \( \frac{2x}{(x-3)(x+3)} = \frac{10}{x^2 - 9} \). Multiply both sides by \(x^2 - 9\) to clear the fractions. This gives \(2x = 10\). Solve for x by dividing both sides by 2, giving \(x = 5\).
03

Check your solution

Substitute x = 5 back into the original equation to check the solution. Substitution gives \(\frac{1}{5-3} + \frac{1}{5+3} = \frac{10}{5^2 - 9}\), which simplifies to \(\frac{1}{2} + \frac{1}{8} = \frac{10}{16}\). Both sides simplify to \(\frac{5}{8}\), confirming the solution is correct.

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{3 x+4}{5} $$

Writing In your own words, explain the meanings of domain and range.

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)= \begin{cases}-x, & x \leq 0 \\ x^{2}-3 x, & x>0\end{cases} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{5}-2 $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{4}{x} $$
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