Question

Solve the equation. Remember to check for extraneous solutions. $$\frac{4}{x(x+1)}=\frac{3}{x}$$

Short answer

The solution to the given equation is \(x=\frac{1}{3}\).

Step-by-step solution

Cross Multiplication

In order to avoid the fractions, apply cross multiplication. So you will multiply each side of the equation by \(x\) and by \(x(x+1)\). Resulting in \(4x=3x(x+1)\)

Simplify and Rearrange the Equation

Now, simplify the right side of equation by distributing 3x, you get \(4x=3x^2+3x\). To simplify, subtract 4x from each side to set the equation equal to zero resulting in \(3x^2+3x-4x=0\) which simplifies to \(3x^2-x=0\).

Factor the Equation

To solve the quadratic equation, we need to factor it first. This simplifies to \(x(3x-1)=0\).

Solve for x

The value of x is then obtained by setting each factor equal to zero and solving: \(x=0\) and \((3x-1)=0\) resulting in \(x=0\) and \(x=\frac{1}{3}\).

Check for Extraneous Solutions

Substitute each solution back into the original equation: with \(x=0\), the original equation becomes undefined because it will result in dividing by zero. So, \(x=0\) is an extraneous solution. With \(x=\frac{1}{3}\), the left side of the original equation equals the right, meaning it is a valid solution.