Question

question
solve for all values of x.
\\(\frac{x + 4}{x + 6} = \frac{-1}{x}\\)
answer attempt 1 out of 3

Explanation:

Step1: Cross - multiply to eliminate fractions

Cross - multiplying the equation \(\frac{x + 4}{x+6}=\frac{-1}{x}\) gives us \(x(x + 4)=-1(x + 6)\).

Step2: Expand both sides

Expanding the left - hand side: \(x^{2}+4x\), and the right - hand side: \(-x - 6\). So the equation becomes \(x^{2}+4x=-x - 6\).

Step3: Rearrange into standard quadratic form

Add \(x\) and \(6\) to both sides of the equation to get \(x^{2}+4x+x + 6 = 0\), which simplifies to \(x^{2}+5x + 6=0\).

Step4: Factor the quadratic equation

We factor \(x^{2}+5x + 6\) as \((x + 2)(x+3)=0\) (since \(2\times3 = 6\) and \(2 + 3=5\)).

Step5: Solve for x

Using the zero - product property, if \((x + 2)(x + 3)=0\), then either \(x+2 = 0\) or \(x + 3=0\).

• If \(x+2=0\), then \(x=-2\).
• If \(x + 3=0\), then \(x=-3\).

We also need to check for extraneous solutions. The original equation has denominators \(x + 6\) and \(x\), so \(x
eq0\) and \(x
eq - 6\). Both \(x=-2\) and \(x=-3\) do not make the denominators zero.

Answer:

\(x=-2\) or \(x=-3\)