Question

solve for x and simplify the answer fully.
\\(\frac{x - 1}{x + 1} = \frac{9}{7}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{x - 1}{x + 1}=\frac{9}{7}\), we use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So, we have \((x - 1)\times7=(x + 1)\times9\).

Step2: Expand both sides

Expand the left - hand side: \(7(x - 1)=7x-7\). Expand the right - hand side: \(9(x + 1)=9x + 9\). So the equation becomes \(7x-7 = 9x+9\).

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

Subtract \(7x\) from both sides: \(7x-7-7x=9x + 9-7x\), which simplifies to \(-7 = 2x+9\). Then subtract 9 from both sides: \(-7-9=2x+9 - 9\), so \(-16 = 2x\).

Step4: Solve for x

Divide both sides by 2: \(\frac{-16}{2}=\frac{2x}{2}\), which gives \(x=- 8\)? Wait, no, wait. Wait, let's re - do step 3 and 4.

Wait, starting from \(7x-7 = 9x + 9\). Subtract \(9x\) from both sides: \(7x-9x-7=9x-9x + 9\), so \(-2x-7 = 9\). Then add 7 to both sides: \(-2x-7 + 7=9 + 7\), so \(-2x=16\). Then divide both sides by \(-2\): \(x=\frac{16}{-2}=-8\)? Wait, no, that can't be. Wait, let's check the cross - multiplication again.

Wait, the original equation is \(\frac{x - 1}{x + 1}=\frac{9}{7}\). Cross - multiplying: \(7(x - 1)=9(x + 1)\). So \(7x-7 = 9x+9\). Then, subtract \(7x\) from both sides: \(-7=2x + 9\). Then subtract 9: \(-16 = 2x\), so \(x=-8\). Wait, but let's check if this is correct. Plug \(x = - 8\) into the original equation: Left - hand side: \(\frac{-8 - 1}{-8+1}=\frac{-9}{-7}=\frac{9}{7}\), which is equal to the right - hand side. So it is correct.

Wait, but I think I made a mistake in the sign when moving terms. Let's do it again.

Starting from \(7x-7 = 9x + 9\). Subtract \(9x\) from both sides: \(7x-9x-7=9\), so \(-2x-7 = 9\). Then add 7: \(-2x=16\), so \(x=-8\). Yes, that's correct.

Answer:

\(x = - 8\)