`1lt2xlt2`
`{:("Quantity A","Quantity B"),(x^(5),x^(7)):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the inequality \(1 < 2x < 2\) and compare the quantities \(x^5\) and \(x^7\), we can follow these steps:

### Step 1: Solve the inequality
We start with the inequality:
\[ 1 < 2x < 2 \]
We can break this down into two parts.

1. From \(1 < 2x\):
\[ \frac{1}{2} < x \]

2. From \(2x < 2\):
\[ x < 1 \]

Combining these results, we have:
\[ \frac{1}{2} < x < 1 \]

### Step 2: Choose a value for \(x\)
To compare \(x^5\) and \(x^7\), we can choose a value of \(x\) that lies within the interval \(\left(\frac{1}{2}, 1\right)\). A suitable choice is:
\[ x = \frac{3}{4} \]

### Step 3: Calculate \(x^5\) and \(x^7\)
Now we calculate \(x^5\) and \(x^7\) using \(x = \frac{3}{4}\).

1. Calculate \(x^5\):
\[ x^5 = \left(\frac{3}{4}\right)^5 = \frac{3^5}{4^5} = \frac{243}{1024} \]

2. Calculate \(x^7\):
\[ x^7 = \left(\frac{3}{4}\right)^7 = \frac{3^7}{4^7} = \frac{2187}{16384} \]

### Step 4: Compare \(x^5\) and \(x^7\)
We know that for \(0 < x < 1\), \(x^n\) decreases as \(n\) increases. Therefore, since \(5 < 7\):
\[ x^5 > x^7 \]

### Conclusion
Thus, we conclude that:
\[ \text{Quantity A } (x^5) > \text{ Quantity B } (x^7) \]

### Final Answer
The relationship is:
\[ \text{Quantity A} > \text{Quantity B} \]

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