To determine which expression is equivalent to [tex]\(\frac{1}{2} x + (-7) - 2 \frac{1}{4} x - (-2)\)[/tex], we need to follow a series of simplification steps. Here's a detailed, step-by-step solution:
1. Rewrite Mixed Fractions as Improper Fractions:
- The term [tex]\(2 \frac{1}{4} x\)[/tex] can be rewritten as [tex]\(\frac{9}{4} x\)[/tex], since [tex]\(2 \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}\)[/tex].
2. Rewrite the Expression:
- Substituting the improper fraction back into the expression, we get:
[tex]\[
\frac{1}{2} x + (-7) - \frac{9}{4} x - (-2)
\][/tex]
3. Simplify Negative Signs:
- Distribute the negative signs:
[tex]\[
\frac{1}{2} x - 7 - \frac{9}{4} x + 2
\][/tex]
4. Combine Like Terms:
- First, combine the [tex]\(x\)[/tex] terms. To do this, we need a common denominator:
[tex]\[
\frac{1}{2} x = \frac{2}{4} x \quad \text{so the x terms are} \quad \frac{2}{4} x - \frac{9}{4} x = \frac{2 - 9}{4} x = -\frac{7}{4} x
\][/tex]
- Next, combine the constant terms:
[tex]\[
-7 + 2 = -5
\][/tex]
5. Construct the Final Expression:
- Combining the simplified [tex]\(x\)[/tex] term and the constant term, we get:
[tex]\[
-\frac{7}{4} x - 5
\][/tex]
- Converting [tex]\(-\frac{7}{4}\)[/tex] to a mixed fraction yields [tex]\(-1 \frac{3}{4}\)[/tex].
Therefore, the expression equivalent to [tex]\(\frac{1}{2} x + (-7) - 2 \frac{1}{4} x - (-2)\)[/tex] is:
[tex]\[
-\frac{7}{4} x - 5 \quad \text{or} \quad -1 \frac{3}{4} x - 5
\][/tex]
Thus, the correct choice is:
(A) [tex]\(-1 \frac{3}{4} x - 5\)[/tex]
