Let's simplify the given expression step-by-step:
We start with the expression:
[tex]\[
-5 \frac{1}{4} - \left( -7 \frac{1}{2} \right)
\][/tex]
First, let's convert the mixed numbers to improper fractions.
For [tex]\(-5 \frac{1}{4}\)[/tex]:
[tex]\[
-5 \frac{1}{4} = -\left(5 + \frac{1}{4}\right) = -\left(\frac{20}{4} + \frac{1}{4}\right) = -\frac{21}{4}
\][/tex]
For [tex]\(-7 \frac{1}{2}\)[/tex]:
[tex]\[
-7 \frac{1}{2} = -\left(7 + \frac{1}{2}\right) = -\left(\frac{14}{2} + \frac{1}{2}\right) = -\frac{15}{2}
\][/tex]
Next, we substitute the improper fractions back into the expression:
[tex]\[
-\frac{21}{4} - \left( -\frac{15}{2} \right)
\][/tex]
To subtract, we change the subtraction of a negative to addition:
[tex]\[
-\frac{21}{4} + \frac{15}{2}
\][/tex]
To add these fractions, we need a common denominator. The common denominator for [tex]\(4\)[/tex] and [tex]\(2\)[/tex] is [tex]\(4\)[/tex]. We convert [tex]\(\frac{15}{2}\)[/tex] to a fraction with a denominator of [tex]\(4\)[/tex]:
[tex]\[
\frac{15}{2} = \frac{15 \times 2}{2 \times 2} = \frac{30}{4}
\][/tex]
Now, we can add the fractions:
[tex]\[
-\frac{21}{4} + \frac{30}{4} = \frac{-21 + 30}{4} = \frac{9}{4}
\][/tex]
Finally, let's convert the improper fraction [tex]\(\frac{9}{4}\)[/tex] to a mixed number:
[tex]\[
\frac{9}{4} = 2 \frac{1}{4}
\][/tex]
Therefore, the simplified result of the expression is:
[tex]\[
2 \frac{1}{4}
\][/tex]
