To ensure that the quotient of [tex]\(\frac{1}{4}\)[/tex] divided by a number is greater than [tex]\(\frac{1}{4}\)[/tex], we need to analyze the inequality:
[tex]\[
\frac{\frac{1}{4}}{x} > \frac{1}{4}
\][/tex]
Simplify the given inequality by multiplying both sides by [tex]\(4\)[/tex]:
[tex]\[
\frac{1}{4} \cdot \frac{1}{x} > \frac{1}{4}
\][/tex]
[tex]\[
\frac{1}{x} > 1
\][/tex]
To find [tex]\(x\)[/tex], we now need to solve this inequality. Multiplying both sides by [tex]\(x\)[/tex] and assuming [tex]\(x\)[/tex] is positive (as we are dividing by this number), we get:
[tex]\[
1 > x
\][/tex]
This simplifies our solution.
Therefore, the number [tex]\(x\)[/tex] should be:
[tex]\[
0 < x < 1
\][/tex]
As a result, any number [tex]\(x\)[/tex] in the range [tex]\(0 < x < 1\)[/tex] will satisfy the given condition that [tex]\(\frac{1}{4}\)[/tex] divided by [tex]\(x\)[/tex] will be greater than [tex]\(\frac{1}{4}\)[/tex].
For an example, if we choose [tex]\(x = 0.5\)[/tex], then:
[tex]\[
\frac{\frac{1}{4}}{0.5} = \frac{1}{4} \cdot \frac{1}{0.5} = \frac{1}{4} \cdot 2 = \frac{1}{2}
\][/tex]
Since [tex]\(\frac{1}{2} > \frac{1}{4}\)[/tex], our condition holds true.
