To determine which of the given options is equivalent to the expression [tex]\(\frac{p+7}{7} - \frac{p-7}{7}\)[/tex], let's simplify the expression step-by-step.
1. Write the given expression:
[tex]\[
\frac{p+7}{7} - \frac{p-7}{7}
\][/tex]
2. Combine the terms over a common denominator:
Since both terms already have the same denominator (7), we can combine them directly:
[tex]\[
\frac{(p+7) - (p-7)}{7}
\][/tex]
3. Distribute and simplify the numerator:
Simplify the expression inside the numerator:
[tex]\[
(p + 7) - (p - 7)
\][/tex]
This becomes:
[tex]\[
p + 7 - p + 7
\][/tex]
Combine like terms:
[tex]\[
p - p + 7 + 7 = 0 + 14 = 14
\][/tex]
So, the simplified numerator is 14.
4. Write the simplified expression with the new numerator:
[tex]\[
\frac{14}{7}
\][/tex]
5. Simplify the fraction:
[tex]\[
\frac{14}{7} = 2
\][/tex]
Given this simplification, we need to check which option matches the value 2.
- (A) [tex]\(\frac{p - p}{7} = \frac{0}{7} = 0\)[/tex]
- (B) [tex]\(\frac{p + p}{7} = \frac{2p}{7}\)[/tex] (This is not a constant and does not equal 2)
- (C) [tex]\(\frac{7 + 7}{7} = \frac{14}{7} = 2\)[/tex]
- (D) [tex]\(\frac{7 - 7}{7} = \frac{0}{7} = 0\)[/tex]
The option that equals 2 is (C).
Thus, the equivalent expression to [tex]\(\frac{p+7}{7} - \frac{p-7}{7}\)[/tex] is:
(C) [tex]\(\frac{7+7}{7}\)[/tex]
