Let's solve the problem step by step to determine the number of thirty-minute sessions Carly had, denoted by [tex]\(x\)[/tex].
1. Understand the given information:
- Carly earns \[tex]$15 for each thirty-minute session.
- Carly earns \$[/tex]25 for each sixty-minute session.
- She earned a total of \$230 over the weekend.
- The number of thirty-minute sessions is [tex]\(x\)[/tex].
- The number of sixty-minute sessions is [tex]\(x - 2\)[/tex].
2. Set up the equation:
- The total earnings from the thirty-minute sessions are [tex]\(15x\)[/tex] dollars.
- The total earnings from the sixty-minute sessions are [tex]\(25(x - 2)\)[/tex] dollars.
- The total earnings combine to give the equation:
[tex]\[
15x + 25(x - 2) = 230
\][/tex]
3. Simplify the equation:
[tex]\[
15x + 25(x - 2) = 230
\][/tex]
Distribute the 25 in the second term:
[tex]\[
15x + 25x - 50 = 230
\][/tex]
Combine like terms:
[tex]\[
40x - 50 = 230
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Add 50 to both sides of the equation to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
40x - 50 + 50 = 230 + 50
\][/tex]
[tex]\[
40x = 280
\][/tex]
Divide both sides by 40 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{280}{40}
\][/tex]
[tex]\[
x = 7
\][/tex]
Therefore, Carly had [tex]\(7\)[/tex] thirty-minute sessions.
The correct answer is [tex]\( \boxed{7} \)[/tex].
