To determine which expression among the given ones has the greatest value, let's evaluate each of them step-by-step using example values for [tex]\( p \)[/tex] and [tex]\( n \)[/tex] that meet the conditions: [tex]\( p \)[/tex] is positive, [tex]\( n \)[/tex] is negative, and [tex]\( |p| > |n| \)[/tex].
Let [tex]\( p = 10 \)[/tex] and [tex]\( n = -4 \)[/tex]. We will evaluate each expression:
1. [tex]\( F: \left|\frac{p - n}{p}\right| \)[/tex]
Substituting the values:
[tex]\[
F = \left|\frac{10 - (-4)}{10}\right| = \left|\frac{10 + 4}{10}\right| = \left|\frac{14}{10}\right| = 1.4
\][/tex]
2. [tex]\( G: \left|\frac{p - n}{n}\right| \)[/tex]
Substituting the values:
[tex]\[
G = \left|\frac{10 - (-4)}{-4}\right| = \left|\frac{10 + 4}{-4}\right| = \left|\frac{14}{-4}\right| = \left| -3.5 \right| = 3.5
\][/tex]
3. [tex]\( H: \left|\frac{p + n}{p - n}\right| \)[/tex]
Substituting the values:
[tex]\[
H = \left|\frac{10 + (-4)}{10 - (-4)}\right| = \left|\frac{10 - 4}{10 + 4}\right| = \left|\frac{6}{14}\right| = \left|\frac{3}{7}\right| \approx 0.42857
\][/tex]
4. [tex]\( J: \left|\frac{p + n}{p}\right| \)[/tex]
Substituting the values:
[tex]\[
J = \left|\frac{10 + (-4)}{10}\right| = \left|\frac{10 - 4}{10}\right| = \left|\frac{6}{10}\right| = 0.6
\][/tex]
5. [tex]\( K: \left|\frac{p + n}{n}\right| \)[/tex]
Substituting the values:
[tex]\[
K = \left|\frac{10 + (-4)}{-4}\right| = \left|\frac{10 - 4}{-4}\right| = \left|\frac{6}{-4}\right| = \left| -1.5 \right| = 1.5
\][/tex]
Now we compare the evaluated results:
- [tex]\( F = 1.4 \)[/tex]
- [tex]\( G = 3.5 \)[/tex]
- [tex]\( H \approx 0.42857 \)[/tex]
- [tex]\( J = 0.6 \)[/tex]
- [tex]\( K = 1.5 \)[/tex]
Among these values, the greatest value is [tex]\( G = 3.5 \)[/tex].
Therefore, the expression [tex]\( \left|\frac{p - n}{n}\right| \)[/tex] has the greatest value. Hence, the answer is:
[tex]\[
\boxed{\text{G}}
\][/tex]
