Question

Solve the equation.
StartFraction x plus 5 Over 5 EndFraction equals 2 minus StartFraction x plus 4 Over 3 EndFraction
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Part 1
What is the​ solution? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet
  
. ​(Type an integer or a​ fraction.)
B.
The solution set is StartSet x font size decreased by 3 | font size decreased by 3 x is a real number EndSet.
C.
The solution set is empty set.

Answer 1

To solve the equation

\[
\frac{x + 5}{5} = 2 - \frac{x + 4}{3},
\]

we begin by eliminating the fractions by finding a common denominator. The denominators are 5 and 3, and the least common multiple is 15. We will multiply each term by 15:

\[
15 \cdot \frac{x + 5}{5} = 15 \cdot \left(2 - \frac{x + 4}{3}\right).
\]

This simplifies to:

\[
3(x + 5) = 30 - 5(x + 4).
\]

Expanding both sides, we get:

\[
3x + 15 = 30 - 5x - 20.
\]

This further simplifies to:

\[
3x + 15 = 10 - 5x.
\]

Next, we will get all the \(x\) terms on one side and the constant terms on the other side. Adding \(5x\) to both sides results in:

\[
3x + 5x + 15 = 10,
\]

which simplifies to:

\[
8x + 15 = 10.
\]

Now, we subtract 15 from both sides:

\[
8x = 10 - 15,
\]

which gives us:

\[
8x = -5.
\]

Finally, we divide by 8:

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

Now let's check if there are any restrictions on the solution. Since there are no divisions by zero and both sides are defined for any real number \(x\), we conclude that the solution set contains this value.

The correct choice is:

A. The solution set is \(\{-\frac{5}{8}\}\).