To determine which expression is equivalent to [tex]\(\frac{1}{2} x + (-7) - 2 \frac{1}{4} x - (-2)\)[/tex], we will simplify the given expression step-by-step.
1. Start with the given expression:
[tex]\[
\frac{1}{2} x + (-7) - 2 \cdot \frac{1}{4} x - (-2)
\][/tex]
2. Simplify each part of the expression individually. First, simplify the multiplication:
[tex]\[
2 \cdot \frac{1}{4} x = \frac{1}{2} x
\][/tex]
Therefore, we substitute back into the expression:
[tex]\[
\frac{1}{2} x + (-7) - \frac{1}{2} x - (-2)
\][/tex]
3. Simplify the subtraction of the fractions involving [tex]\(x\)[/tex]. Subtract [tex]\(\frac{1}{2} x\)[/tex] from [tex]\(\frac{1}{2} x\)[/tex]:
[tex]\[
\frac{1}{2} x - \frac{1}{2} x = 0
\][/tex]
Thus, the expression is simplified to:
[tex]\[
-7 - (-2)
\][/tex]
4. Simplify the remaining constants:
[tex]\[
-7 - (-2) = -7 + 2 = -5
\][/tex]
Therefore, the equivalent simplified expression is:
[tex]\[
-5
\][/tex]
Now, among the given multiple choices:
A) [tex]\(-1 \frac{3}{4} x - 5\)[/tex] \\
B) [tex]\(1 \frac{3}{4} x - 9\)[/tex] \\
C) [tex]\(3 \frac{3}{4} x - 9\)[/tex] \\
D) [tex]\(3 \frac{3}{4} x - 7\)[/tex]
The correct expression equivalent to [tex]\(\frac{1}{2} x + (-7) - 2 \frac{1}{4} x - (-2)\)[/tex] is indeed:
[tex]\[
-5
\][/tex]
Thus, the index of the correct option corresponding to [tex]\(-1 \frac{3}{4} x - 5\)[/tex] (representing [tex]\(\frac{-7x}{4} - 5\)[/tex]), which simplifies to [tex]\(-5\)[/tex], is option A.
