Thread: Inequalities with variable in denominator

I have a question about inequalities like 1/x < 4. In this case the answer is (neg infinity, 0) U (1/4, pos infinity). It's intuitive that the set of negatives should be included, but the mechanical process of solving for x just goes out the window. That is, what rule do I break if i were to just solve for x... x>1/4 so the set would be simply (1/4, pos infinity). And, how do i know when the solving for x alone will not work.

Originally Posted by eniuqvw

I have a question about inequalities like 1/x < 4. In this case the answer is (neg infinity, 0) U (1/4, pos infinity). It's intuitive that the set of negatives should be included, but the mechanical process of solving for x just goes out the window. That is, what rule do I break if i were to just solve for x... x>1/4 so the set would be simply (1/4, pos infinity). And, how do i know when the solving for x alone will not work.

Hello,

Ok, there is one thing you should know. When multiplying by a negative number, the direction of the inequality changes.

For example : 2<3
When multiplying by -1, you get -2>-3


So here, intuitively, you should separate the situations :
1) firstly, if x is positive
2) secondly, if x is negative


1) If x is positive :

1/x<4

Multiplying by x, we get : 1<4x, since x>0.
--> x>1/4

Solution to this is (1/4, pos infinity)

2) If x is negative :

Here, it's immediate because 1/x is negative, hence always inferior to 4.
But let's apply the method.

1/x<4

Multiplying by x, we get 1>4x, because x is negative !

--> x<1/4

Because x is negative, you gotta find all the negative numbers which verify x<1/4.
This is true for all x negative.

Thus the solution to this is (neg infinity, 0)

Originally Posted by eniuqvw

I have a question about inequalities like 1/x < 4. In this case the answer is (neg infinity, 0) U (1/4, pos infinity). It's intuitive that the set of negatives should be included, but the mechanical process of solving for x just goes out the window. That is, what rule do I break if i were to just solve for x... x>1/4 so the set would be simply (1/4, pos infinity). And, how do i know when the solving for x alone will not work.

Because my algebra is so poor I always solve inequalities by drawing graphs and then solving an equality.

In your case I'd draw the graphs of y = 1/x and y = 1/4. Then I'd solve 1/x = 4. Then I'd use the graphs to get the solution to the inequality.

Originally Posted by eniuqvw

I have a question about inequalities like 1/x < 4. In this case the answer is (neg infinity, 0) U (1/4, pos infinity).

First Method: make the RHS,



To get a negative quantity we must study two cases:

• If & Intersect these conditions and we have
  
  
• If & Intersect these conditions and we have

Final solution we'll be given by


Second Method: we obviously discard otherwise the initial inequality makes no sense. So we're gonna multiply both sides by since this quantity is positive, the inequality stays as it's written. Following up on this we have:

From here study two cases again:

• If &
  
  
• If &

Hence which is we got before.

A much easier approach is to multiply both sides by x^2 instead of x.

As x^2 is always greater than 0 (x = 0 is not allowed in this case), there is no need to worry about the direction of the inequality sign.

For 1/x < 4
Multiplying both sides by x^2, we have
x < 4x^2
4x^2 - x > 0
x(4x - 1) > 0

The two roots of x(x/4 - 1) are x = 0 and x = 1/4.

So the solution to the inequality is
x < 0 or x > 1/4

Then you're not gonna have trouble to check the second method above.