To determine the number of solutions for the equation [tex]\(2(x + 4) - 1 = 2x + 7\)[/tex], let's go through the steps.
1. Distribute the 2 inside the parentheses:
[tex]\[
2(x + 4) - 1 = 2x + 7
\][/tex]
This becomes:
[tex]\[
2x + 8 - 1 = 2x + 7
\][/tex]
2. Simplify the left side:
[tex]\[
2x + 7 = 2x + 7
\][/tex]
3. Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
2x + 7 - 2x = 2x + 7 - 2x
\][/tex]
This results in:
[tex]\[
7 = 7
\][/tex]
At this point, we observe that the equation [tex]\(7 = 7\)[/tex] is always true, regardless of the value of [tex]\(x\)[/tex]. This indicates that the original equation is an identity and holds for all possible values of [tex]\(x\)[/tex].
Therefore, the equation [tex]\(2(x + 4) - 1 = 2x + 7\)[/tex] does not have a specific, finite number of solutions but rather has an infinite number of solutions.
Thus, the correct answer is:
C. infinite
