Question

solve: \\(\frac{16}{x - 2} = \frac{32}{x + 2}\\)\
\\(x = \square\\)

Explanation:

Step1: Cross - multiply the equation

To solve the rational equation \(\frac{16}{x - 2}=\frac{32}{x + 2}\), we use the cross - multiplication property. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So, we have \(16\times(x + 2)=32\times(x - 2)\).

Step2: Expand both sides

Expand the left - hand side: \(16x+32\), and the right - hand side: \(32x-64\). So the equation becomes \(16x + 32=32x-64\).

Step3: Move the x - terms to one side and constants to the other

Subtract \(16x\) from both sides: \(16x+32 - 16x=32x-64 - 16x\), which simplifies to \(32 = 16x-64\). Then add 64 to both sides: \(32 + 64=16x-64 + 64\), so \(96 = 16x\).

Step4: Solve for x

Divide both sides by 16: \(x=\frac{96}{16}=6\). We also need to check if \(x = 6\) makes the denominators non - zero. When \(x = 6\), \(x-2=6 - 2 = 4
eq0\) and \(x + 2=6 + 2 = 8
eq0\).

Answer:

\(6\)