So, I have the equation \({x}^{2} < 1\) and I'm trying to isolate x. I believe the solution is \(-1 < x < 1\), but I can't seem to isolate x properly to figure it out.
+23251 There are several ways to work this problem; this is one way:
x2 < 1
Subtract 1 from both sides: x2 - 1 < 0
Factor: (x + 1)(x - 1) < 0
Analysis: If two numbers are multiplied together and their answer is negative (less than 0), then either the first number is
positive and the second number is negative or the first number is negative and the second number is positive.
For this problem, the two numbers are x + 1 and x - 1.
So: first number positive, second number negative or first number negative, second number positive
(x + 1) > 0 and (x - 1) < 0 (x + 1) < 0 and (x - 1) > 0
x + 1 > 0 and x - 1 < 0 x + 1 < 0 and x - 1 > 0
x > -1 and x < 1 x < -1 and x > 1
-1 < x < 1 <this is impossible>
Thus: -1 < x < 1
+23251 There are several ways to work this problem; this is one way:
x2 < 1
Subtract 1 from both sides: x2 - 1 < 0
Factor: (x + 1)(x - 1) < 0
Analysis: If two numbers are multiplied together and their answer is negative (less than 0), then either the first number is
positive and the second number is negative or the first number is negative and the second number is positive.
For this problem, the two numbers are x + 1 and x - 1.
So: first number positive, second number negative or first number negative, second number positive
(x + 1) > 0 and (x - 1) < 0 (x + 1) < 0 and (x - 1) > 0
x + 1 > 0 and x - 1 < 0 x + 1 < 0 and x - 1 > 0
x > -1 and x < 1 x < -1 and x > 1
-1 < x < 1 <this is impossible>
Thus: -1 < x < 1
Thanks, geno3141.
I actually managed to figure this out (in the exact same way) while I was going out to do errands, but thanks for verifying the solution.
I am curious about the case where x < -1 and x > 1. Obviously this is a contradiction, but what does it mean? How can this contradiction be resolved? Or does it not need to be resolved, and it suffices to only have one half of the method provide a relevant answer?
+129840 Think about the problem in this way:
If we square any positive real number greater or equal to 1, we will always get a result that is greater or equal to 1
Likewise....{and this is tougher to see} ......, squaring any negative real number that less than or equal to -1 will also give a result that is greater or equal to 1......example (-1.1)^2 = 1.21
This is what geno is trying to say.......that x cannot come from either of the intervals x ≥ 1 or x ≤ -1
Thus.....x must come from the interval (-1, 1)
What I am asking is a more general question though: why does a contradiction arise when using the method?
It's quite obvious that squaring a number greater than 1 or less than its opposite -1 will be greater than 1. I understand that .
My question is: why do we even have one answer which is wrong (contradictory) from this method? Is it a problem of the method, a problem with the domain of the question, or not a problem at all, and just a contradiction to be rationally discarded because it's not useful?
Does the contradiction simply mean that the premises (x + 1) < 0 and (x - 1) > 0 were not correct, and the other premises (x + 1) > 0 and (x - 1) < 0 were?
I want to be generally aware of how I should approach math problems.
