SOLUTION: Solve this inequalities 1/x2 <= 1 Thanks Tutor :)

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Question 230545: Solve this inequalities
1/x2 <= 1
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Found 2 solutions by jsmallt9, Theo:
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!

Let's start by eliminating the fraction. Multiply both sides by . (Since is never negative we do not have to be concerned about reversing the inequality.)

Since this is a quadratic. let's get one side equal to zero by subtracting 1 from each side:

and then factoring:

This tells us that the product on the right is zero or positive.
Let's think about positive products. When you multiply two factors and you get a positive result, what must be true about the factors? Answer: They must both be positive or both be negative. And how do say this, in the form of Mathematical sentences (equations or inequalities)? Here are some key ideas to a simple answer:
No matter what number x may be, (x+1) will always be larger than (x-1).
If the smaller of two numbers is positive, then the larger one would have to be positive, too.
If the larger of two numbers is negative, then the smaller one would also have to be negative.

Putting these three ideas together we can say the two factors are positive with and say that both are negative with . (The "or equal to" part of these inequalities handles the products that are zero which we also can accept.)
So now we have:
or .
Solving these we get:
or

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
I believe the formula you are looking to solve is:

If so, then the solution would appear to be as follows:

multiply both sides of the equation by x^2 to get:

1 <= x^2

This is the same as x^2 >= 1

Take the square root of both sides of this equation to get:

x >= 1
or:
x <= -1

How I got to this part I can't really explain that well, but it works.

square root of x^2 is x.

No problem there, because x * x = x^2 regardless if x is positive or negative.

If it was = rather than >=, the answer would have been:

x = +/- 1

Because it was >=, then you have to take into account that when you multiply both sides of an inequality by -1, then the inequality reverses.

This causes:

x >= 1
or
x <= -1

To confirm the answer is good, you would take some values of x and replace x in the original equation with them.
Take values in and out of the acceptable range.

Let x equal:
-2,-1,-.5,0,.5,1,2

Then 1/x^2 <= 1 becomes:

When x = -2, 1/x^2 = 1/4 < 1 = ok.
When x = -1, 1/x^2 = 1/1 = 1 = ok
When x = -.5, 1/x^2 = 1/.25 > 1 = NOT ok.
When x = 0, 1/x^2 = 1/0 = undefined = NOT ok.
When x = .5, 1/x^2 = 1/.25 > 1 = NOT ok.
When x = 1, 1/x^2 = 1/1 = 1 = ok.
When x = 2, 1/x^2 = 1/4 < 1 = ok.

Your answer is that x >= 1 or x <= -1.

You do not have to restrict the domain to eliminate division by 0 because the domain already excludes 0.