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Which expression is equivalent to [tex]\frac{1}{2} x + (-7) - 2 \frac{1}{4} x - (-2)?[/tex]
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]
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]
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