Nups1324 wrote:
Nups1324 wrote:
Is x > 0?
(1) 1/x < 1
(2) 1/x > x
Source: Experts' GlobalHi
Maxximus I checked the video explanation but I have a doubt.
Why can't we multiply by x on both sides and thus get 1<x from statement 1 and 1>x^2 from statement.? Why do we need to only test numbers as shown in the video explanation?
Thank you
First of all, testing numbers is not the only way to solve this question.
The reason we wouldn’t multiply each side of 1/x < 1 by x is that we have no information on the sign of x. If x were positive, then we would obtain x > 1, as you noticed, but if x were negative, we would obtain x < 1 since we have to change the direction of an inequality whenever we multiply each side by a negative number.
To solve 1/x < 1, move the 1 on the right hand side to the left hand side:
1/x - 1 < 0
Now get the common denominator x and then combine the two fractions:
1/x - x/x < 0
(1 - x)/x < 0
Now, there are two ways this expression can be negative:
Case 1: 1 - x is negative and x is positive, or
Case 2: 1 - x is positive and x is negative.
For
Case 1, we get 1 - x < 0 (which is equivalent to x > 1) and x > 0. The set of numbers that satisfy both of the inequalities x > 0 and x > 1 are x > 1. So, one way to satisfy the inequality 1/x < 1 is to pick values for x which are greater than 1.
For
Case 2, we get 1 - x > 0 (which is equivalent to x < 1) and x < 0. The set of numbers that satisfy both inequalities are the set of x such that x < 0. Thus, the inequality 1/x < 1 can also be satisfied by picking a negative value for x.
As we can see, x could be either greater than 1 or less than 0. If we were to multiply each side of 1/x < 1 by x, we would completely miss the fact that negative values of x also satisfy the inequality 1/x < 1.
The explanation of why it’s wrong to rewrite 1/x > x as x^2 < 1 is similar to the explanation just presented.