Given 1<x<2.
Which is the correct relationship between quantities P and Q based on the given information?
| P | Q |
|---|---|
| 1−4x1−16x | 1−4−x1−16−x |
Explanation
We will simplify the forms of P and Q first.
Simplifying P
Recall that a2−b2=(a−b)(a+b). Note that 16x=(4x)2.
P=1−4x1−16x=1−4x1−(4x)2
P=1−4x(1−4x)(1+4x)
P=1+4x
Simplifying Q
Similarly, note that 16−x=(4−x)2.
Q=1−4−x1−16−x=1−4−x1−(4−x)2
Q=1−4−x(1−4−x)(1+4−x)
Q=1+4−x
Analyzing the Value of P
Given 1<x<2.
41<4x<42
4<4x<16
1+4<1+4x<1+16
5<P<17
Analyzing the Value of Q
Given 1<x<2. Multiply by −1 (inequality signs flip).
−2<−x<−1
4−2<4−x<4−1
161<4−x<41
1+161<1+4−x<1+41
1617<Q<45
Comparing P and Q
From the results above, we get:
- P>5
- Q<45=1.25
It is clearly visible that the value of P is always greater than Q for the given range of x.
So, P>Q.