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Chapter 0: Problem 44

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{6}{x}-\frac{2}{x+3}=\frac{3(x+5)}{x^{2}+3 x} $$

Short Answer

Expert verified
The solution to the equation is \(x = 3\).

Step by step solution

01

Simplify the Equation

The equation \(\frac{6}{x}-\frac{2}{x+3}=\frac{3(x+5)}{x^{2}+3 x}\) simplifies to \(\frac{6(x + 3) - 2x}{x(x+3)} = \frac{3(x+5)}{x (x + 3)}\).
02

Clear the Fractions

Multiply every term by \(x(x+3)\). This gives \(6x + 18 - 2x^2 = 3(x + 5)\). Rearrange this to give a quadratic equation: \(2x^2 + 3x - 18 = 0\).
03

Solve the Quadratic Equation

We factor the quadratic equation to get \(2(x - 3)(x + 3) = 0\), which gives two roots: \(x = 3\) and \(x = -3\).
04

Check the Potential Solutions

Every solution must be checked in the original equation to ensure it does not make the denominator zero. If \(x = 3\), then the denominators become \(3\) and \(6\) respectively, which are valid values. If \(x = -3\), the denominators do not exist, because it would result in a zero denominator in the original equation. Therefore, \(x = -3\) is an extraneous solution and can not be a solution to the equation.
05

Answer

The only valid solution to the equation is \(x = 3\) after checking them against the original equation.

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Most popular questions from this chapter

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ (g \circ f)^{-1} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=x^{2}+12 x-4 & \quad x_{1}=1, x_{2}=5 \end{array} $$

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 \(f^{-1}\). $$ f(x)=\frac{8 x-4}{2 x+6} $$

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ g^{-1} \cdot f^{-1} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-2 x+15 & x_{1}=0, x_{2}=3 \end{array} $$
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