To determine the value of [tex]\( x \)[/tex], we start by interpreting the given information.
Carly earns [tex]\(\$15\)[/tex] for each thirty-minute session and [tex]\(\$25\)[/tex] for each sixty-minute session. The total amount she earned over the past weekend is [tex]\(\$230\)[/tex]. Additionally, she held [tex]\( x \)[/tex] thirty-minute sessions and [tex]\( x-2 \)[/tex] sixty-minute sessions.
Given this information, we can set up an equation based on her earnings from both types of sessions. The equation representing her total earnings is:
[tex]\[
15x + 25(x - 2) = 230
\][/tex]
First, we distribute the 25 in the second term:
[tex]\[
15x + 25x - 50 = 230
\][/tex]
Next, we combine like terms:
[tex]\[
40x - 50 = 230
\][/tex]
Next, we need to isolate [tex]\( x \)[/tex]. Begin by adding 50 to both sides of the equation:
[tex]\[
40x - 50 + 50 = 230 + 50
\][/tex]
[tex]\[
40x = 280
\][/tex]
Now, divide both sides of the equation by 40 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{280}{40} = 7
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 7 \)[/tex].
Therefore, the correct answer is
[tex]\[
\boxed{7}
\][/tex]
