Question

solve for x. if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions, enter none.
\\(\frac{10}{2x + 8} = \frac{-5}{6x + 5}\\)

Explanation:

Step1: Cross - multiply the fractions

Given the equation \(\frac{10}{2x + 8}=\frac{-5}{6x+5}\), cross - multiplying gives us \(10\times(6x + 5)=-5\times(2x + 8)\).

Step2: Expand both sides

Expanding the left - hand side: \(10\times(6x + 5)=60x+50\).
Expanding the right - hand side: \(-5\times(2x + 8)=-10x-40\).
So the equation becomes \(60x + 50=-10x-40\).

Step3: Add \(10x\) to both sides

Adding \(10x\) to both sides of the equation \(60x + 50=-10x-40\) gives \(60x+10x + 50=-10x + 10x-40\), which simplifies to \(70x+50=-40\).

Step4: Subtract 50 from both sides

Subtracting 50 from both sides: \(70x+50 - 50=-40 - 50\), so \(70x=-90\).

Step5: Solve for \(x\)

Divide both sides by 70: \(x=\frac{-90}{70}=-\frac{9}{7}\).
We need to check if this solution makes the denominators zero.
For the denominator \(2x + 8\), when \(x =-\frac{9}{7}\), \(2\times(-\frac{9}{7})+8=-\frac{18}{7}+\frac{56}{7}=\frac{38}{7}
eq0\).
For the denominator \(6x + 5\), when \(x =-\frac{9}{7}\), \(6\times(-\frac{9}{7})+5=-\frac{54}{7}+\frac{35}{7}=-\frac{19}{7}
eq0\). So \(x =-\frac{9}{7}\) is a valid solution.

Answer:

\(-\frac{9}{7}\)