Let's break down the problem step-by-step:
1. Identify the costs and the total earnings:
- Carly earns [tex]$15 for each thirty-minute session.
- Carly earns $[/tex]25 for each sixty-minute session.
- The total earnings over the weekend are $230.
2. Set up the variables:
- Let [tex]\( x \)[/tex] be the number of thirty-minute sessions.
- According to the problem, Carly had [tex]\( x - 2 \)[/tex] sixty-minute sessions.
3. Write the equation for the total earnings:
- The earnings from thirty-minute sessions would be [tex]\( 15x \)[/tex].
- The earnings from sixty-minute sessions would be [tex]\( 25(x - 2) \)[/tex].
4. Combine these to form the equation:
- The total earnings are given by summing the two parts:
[tex]\[
15x + 25(x - 2) = 230
\][/tex]
5. Simplify and solve the equation:
[tex]\[
15x + 25(x - 2) = 230
\][/tex]
[tex]\[
15x + 25x - 50 = 230
\][/tex]
[tex]\[
40x - 50 = 230
\][/tex]
[tex]\[
40x = 280
\][/tex]
[tex]\[
x = 7
\][/tex]
6. Verify the solution:
- If [tex]\( x = 7 \)[/tex], Carly had 7 thirty-minute sessions.
- The number of sixty-minute sessions would be [tex]\( x - 2 = 7 - 2 = 5 \)[/tex].
- Earnings from thirty-minute sessions: [tex]\( 15 \times 7 = 105 \)[/tex] dollars.
- Earnings from sixty-minute sessions: [tex]\( 25 \times 5 = 125 \)[/tex] dollars.
- Total earnings: [tex]\( 105 + 125 = 230 \)[/tex] dollars, which matches the given total.
Thus, the correct value of [tex]\( x \)[/tex] is
D. 7
