Question

solve the equation:
\\(\frac{12}{x+2} = \frac{9}{x+5}\\)
what are the solutions?
\\(\circ\\) \\(x = 7\\)
\\(\circ\\) \\(x = -14\\) only
\\(\circ\\) \\(x = -1\\)
\\(\circ\\) \\(x = -2, -5\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{12}{x + 2}=\frac{9}{x+5}\), we cross - multiply. Cross - multiplying gives us \(12(x + 5)=9(x + 2)\).

Step2: Expand both sides

Expand the left - hand side: \(12x+60\), and the right - hand side: \(9x + 18\). So the equation becomes \(12x+60 = 9x+18\).

Step3: Subtract \(9x\) from both sides

Subtract \(9x\) from each side: \(12x-9x + 60=9x-9x + 18\), which simplifies to \(3x+60 = 18\).

Step4: Subtract 60 from both sides

Subtract 60 from both sides: \(3x+60 - 60=18 - 60\), so \(3x=-42\).

Step5: Divide both sides by 3

Divide both sides by 3: \(\frac{3x}{3}=\frac{-42}{3}\), which gives \(x=-14\).
We also need to check for extraneous solutions. The original equation has denominators \(x + 2\) and \(x + 5\). When \(x=-14\), \(x + 2=-14 + 2=-12
eq0\) and \(x + 5=-14 + 5=-9
eq0\), so \(x=-14\) is a valid solution.

Answer:

\(x=-14\) only