00:03
In this question, we're asked, in the first part of this question, we're given a function f of x, and we're asked to find the values of x for which the function is positive.
00:16
So we're going to draw a number line, and we're going to, on this number of line, we're going to draw points where each factor in this function is equal to zero.
00:28
So we have three points where factors become zero.
00:33
X equals negative 4 x equals 4 and x equals 5 right these are points where your function is either 0 or undefined meaning you you divide by 0 now to find out when our function is positive so or and when it's negative we need to test so this these three points they split the whole interval into four parts right and we need to test the sign of our function f of x on each of these intervals.
01:14
And to do that, we need to pick a test point on each of the subintervals.
01:21
So, for example, on the first, let's start from the from right and go to the left.
01:28
The first interval, subinterval, is from 5 to infinity.
01:32
On the sub interval 5 to infinity, our function pick any test point, for example, pick x equals to 6, right? and plug in that number in the function.
01:47
The value for function at x equals 6 is equal to 6 minus 5 over 6 plus 4, 6 minus 4.
02:00
And this number is 1 over 10 times 2.
02:05
And we don't really care about the exact value, we just need to know that it's positive.
02:10
This means that on the whole interval from 5 to infinity our function is going to be positive.
02:20
This means f is positive on the interval 5 infinity.
02:27
Now the next interval is from 4 to 5.
02:31
Between 4 and 5 again pick some point in this interval, let's say 4 .5.
02:38
So f of 4 .5 is equal to plug in 4 .5 in the function.
02:46
You will get negative 0 .5 in the numerator you will get 8 .5 in the denominator times 0 .5.
03:04
This is a negative number and this means that the function is going to be negative on the whole interval 4 -5.
03:12
F of x is negative on the interval from 4 to 5...