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Chapter 0: Problem 39

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{2}{(x-4)(x-2)}=\frac{1}{x-4}+\frac{2}{x-2} $$

Short Answer

Expert verified
The solution to the equation includes all real numbers except 4 and 2.

Step by step solution

01

Simplification

Simplify the equation by subtracting \(\frac{1}{x-4}\) and \(\frac{2}{x-2}\) from both sides, this renders the equation to 0 = 0 as the left side and the right side of equation are identical, the original equation is an identity.
02

Identify Domain Restrictions

Recognize that while the equation simplifies to being an identity, there are values which x cannot take. Given the denominators in our original equation, there are two evident restrictions: \(x 鈮� 4\) and \(x 鈮� 2\). These values would make the denominators of our original equation zero, leading to undefined results. Thus, since we subtracted those initial values, while the equation is an identity, these restrictions remain.
03

Concluding the Solution

Since the original equation simplifies to an identity (0=0), the solution will contain all real numbers except for the values identified as restrictions in our second step. That is: \(x 鈭� R, x 鈮� 4 \) and \(x 鈮� 2\).

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Most popular questions from this chapter

College Students The numbers of foreign students \(F\) (in thousands) enrolled in colleges in the United States from 1992 to 2002 can be approximated by the model. $$ F=0.004 t^{4}+0.46 t^{2}+431.6, \quad 2 \leq t \leq 12 $$ where \(t\) represents the year, with \(t=2\) corresponding to 1992. (Source: Institute of International Education) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the context of the problem. (c) Find the five-year time periods when the rate of change was the greatest and the least.

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 \(f^{-1}\). $$ f(x)=x^{3}+1 $$

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ f(x)=\frac{1}{8}(x+2)^{2}-1 $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-\sqrt{x+1}+3 &\quad x_{1}=3, x_{2}=8 \end{array} $$

Your wage is \(\$ 8.00\) per hour plus \(\$ 0.75\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced is $$ y=8+0.75 x $$ (a) Find the inverse function. (b) What does each variable represent in the inverse function? (c) Determine the number of units produced when your hourly wage is \(\$ 22.25\).
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