Question

solve for x and simplify the answer fully.
\\(\frac{5}{5x + 6} = \frac{4}{x + 2}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{5}{5x + 6}=\frac{4}{x + 2}\), we use the cross - multiplication property of fractions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So, we have \(5\times(x + 2)=4\times(5x + 6)\).

Step2: Expand both sides

Expand the left - hand side: \(5\times(x + 2)=5x+10\).
Expand the right - hand side: \(4\times(5x + 6)=20x + 24\).
Now our equation becomes \(5x+10 = 20x+24\).

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

Subtract \(5x\) from both sides: \(5x - 5x+10=20x - 5x+24\), which simplifies to \(10 = 15x+24\).
Then subtract 24 from both sides: \(10-24=15x+24 - 24\), which gives \(- 14 = 15x\).

Step4: Solve for x

Divide both sides by 15: \(x=\frac{-14}{15}\). We need to check if this solution makes the denominators zero.
For the denominator \(5x + 6\): \(5\times(-\frac{14}{15})+6=-\frac{14}{3}+6=\frac{-14 + 18}{3}=\frac{4}{3}
eq0\).
For the denominator \(x + 2\): \(-\frac{14}{15}+2=\frac{-14 + 30}{15}=\frac{16}{15}
eq0\). So, the solution is valid.

Answer:

\(x =-\frac{14}{15}\)