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Chapter 0: Problem 36

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{1}{x}+\frac{2}{x-5}=0 $$

Short Answer

Expert verified
The equation has no real solutions.

Step by step solution

01

Combine terms

To combine \( \frac{1}{x} \) and \( \frac{2}{x-5} \), they need a common denominator. The common denominator of x and x-5 is x(x-5). Multiply each term by this common denominator to get rid of the fractions. This gets you \(x(x-5)\cdot\frac{1}{x} + x(x-5)\cdot\frac{2}{x-5} = 0 \), which simplifies to \(x-5 + 2x = 0\).
02

Simplify equation

Simplify the equation: Combine like terms to get \(3x-5=0\).
03

Solve for x

Solve for x to find \(x=\frac{5}{3}\)
04

Check solution

Substitute \(x=\frac{5}{3}\) back into the original equation to check if the solution is correct. That gives: \( \frac{1}{\frac{5}{3}} + \frac{2}{\frac{5}{3}-5} = 0 \), which simplifies to \( \frac{3}{5} - \frac{2}{-\frac{10}{3}} = 0 \), or \( \frac{3}{5} + \frac{3}{5} = 1 \), not zero, which means \(x=\frac{5}{3}\) is not a solution.
05

Conclusion

There is no solution to the equation because when substituting the value obtained for x back into the original equation, it does not satisfy the equation.

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Most popular questions from this chapter

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

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(t)=t^{2}+2 t-3 $$

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 \(f^{-1}\). $$ f(x)=-\frac{2}{x} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ g(x)=x^{3}-5 x $$
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