Question

Solve for x and simplify the answer fully.
start fraction, x, plus, 5, divided by, 2, x, minus, 6, end fraction, equals, start fraction, 5, divided by, 2, end fraction
2x−6
x+5
​
=
2
5
​

Answer 1

To solve for \(x\) in the equation

\[
\frac{x + 5}{2x - 6} = \frac{5}{2},
\]

we can start by cross-multiplying:

\[
2(x + 5) = 5(2x - 6).
\]

Now distribute on both sides:

\[
2x + 10 = 10x - 30.
\]

Next, we'll move the terms involving \(x\) to one side and the constant terms to the other side. Subtract \(2x\) from both sides:

\[
10 = 10x - 2x - 30.
\]

This simplifies to:

\[
10 = 8x - 30.
\]

Now add \(30\) to both sides:

\[
10 + 30 = 8x,
\]
which simplifies to:

\[
40 = 8x.
\]

Dividing both sides by \(8\) gives:

\[
x = \frac{40}{8} = 5.
\]

So the solution for \(x\) is

\[
\boxed{5}.
\]

Now, let's verify by substituting \(x = 5\) back into the original equation:

The left side becomes:

\[
\frac{5 + 5}{2(5) - 6} = \frac{10}{10 - 6} = \frac{10}{4} = \frac{5}{2}.
\]

The right side is also:

\[
\frac{5}{2}.
\]

Both sides are equal, confirming the solution \(x = 5\) is correct. Thus, the final answer is

\[
\boxed{5}.
\]