Is it Possible for x^2 to Exceed 900?

[mark2142]

Homework Statement

Suppose ##x^2 > 900##

Relevant Equations

What will be x?

We can also write a general inequality ##x^2>a## where a is a number.
If ##x^2= a##, then ##x= \pm \sqrt a## which means ##x= \sqrt a## or ##-\sqrt a##.
But in this case i don’t think it will be ##x > \pm \sqrt a## because if we take ##0## , it’s greater than a negative but in the original inequality ##0 < a## , a positive number.

 

[BvU]

How about ##|x|## ?

 

[PeroK]

If ##x^2= a##, then ##x= \pm \sqrt a## which means ##x= \sqrt a## or ##-\sqrt a##.
But in this case i don’t think it will be ##x > \pm \sqrt a## ...

That's true. ##x^2 > a \Rightarrow x > a## or ##x \ ? -a##?

 

[Steve4Physics]

Homework Statement:: Suppose ##x^2 > 900##
Relevant Equations:: What will be x?

We can also write a general inequality ##x^2>a## where a is a number.
If ##x^2= a##, then ##x= \pm \sqrt a## which means ##x= \sqrt a## or ##-\sqrt a##.
But in this case i don’t think it will be ##x > \pm \sqrt a## because if we take ##0## , it’s greater than a negative but in the original inequality ##0 < a## , a positive number.

##x > \pm \sqrt a## is ambiguous and shouldn’t be used.

For example, if ##a=900## then one possible interpretation of ##x > \pm \sqrt a## is that it is satisfied by ##x=1##, because ##1>-30##, which is clearly wrong does not satisfy the original inequality##x^2 \gt a##.

Edited.

 

[FactChecker]

But in this case i don’t think it will be ##x > \pm \sqrt a## because if we take ##0## , it’s greater than a negative but in the original inequality ##0 < a## , a positive number.

Good catch. When you are dealing with inequalities, you have to be careful about minus signs and negative numbers. You should handle these two cases separately.
Case 1: ##x \gt \sqrt a##
Case 2: ##x \lt -\sqrt a##

 

[BvU]

Dear @mark2142 ,

Are you aware that if ## a > b ##, then it follows that ## - a < - b## ?

I.e. if you multiply left and right of an inequality with a negative number : the inequality sign flips ...

##\ ##

 

[mark2142]

##x > \pm \sqrt a## is ambiguous and shouldn’t be used.

For example, if ##a=900## then one possible interpretation of ##x > \pm \sqrt a## is that it is satisfied by ##x=1##, because ##1>-30##, which is clearly wrong does not satisfy the original inequality##x^2 \gt a##.

Edited.

Yes! ##x> - \sqrt a## is wrong and then ##x^2 > a## is not true.
But how do we reach to the solution?
(I don’t want to solve ##x^2 -a > 0##.)

 

[mark2142]

Good catch. When you are dealing with inequalities, you have to be careful about minus signs and negative numbers. You should handle these two cases separately.
Case 1: ##x \gt \sqrt a##
Case 2: ##x \lt -\sqrt a##

How have you reached to the solution?
( sorry for late response! )

 

[mark2142]

How about ##|x|## ?

What about it?

 

[BvU]

Yes! If ##x> - \sqrt a## then ##x^2 > a## is not true.
But how do we reach to the solution?
(I don’t want to solve ##x^2 -a > 0##.)

Come again ? In post #1 you ask ' what is ##x## if ##x^2>900## ' and now you don't want to solve the equivalent problem ?

How about making a plot, then ?

##\ ##

 

[BvU]

What about it?

##x^2 > 900\ \Leftrightarrow \ |x|>\sqrt{900}##

 

[mark2142]

Come again ? In post #1 you ask ' what is ##x## if ##x^2>900## ' and now you don't want to solve the equivalent problem ?

How about making a plot, then ?

View attachment 323730
##\ ##

I thought maybe there was a fast algebraic method. So tell me about the graph.You plotted a graph between ##x## and ##x^2## by taking different values of x. By graph it’s clear ##x^2> 900## for ##x <-30## and ##x>30##.

 

[BvU]

I thought maybe there was a fast algebraic method

There is:

##x^2 > 900\ \Leftrightarrow \ |x|>\sqrt{900}##

##\Leftrightarrow \ x>30\ \lor \ -x<-30 ##

[edit] Oops ! see #16

 

[FactChecker]

I thought maybe there was a fast algebraic method.

Not really. The best (only?) way is to consider two cases and see where each case takes you.

 

[mark2142]

Not really. The best (only?) way is to consider two cases and see where each case takes you.

Ok! Thanks everyone. Bye!

 

[DrClaude]

##\Leftrightarrow \ x>30\ \lor \ -x<-30 ##

##\Leftrightarrow \ x>30\ \lor \ x<-30 ##

 

[Mark44]

Emphasis added...

So tell me about the graph. You plotted a graph between ##x## and ##x^2## by taking different values of x. By graph it’s clear ##x^2> 900## for ##x <-30## and ##x>30##.

1. @BvU provided a graph of ##y = x^2## and a graph of ##y = 900## to demonstrate visually the intervals for which ##x^2 > 900##.
2. The appropriate conjunction above is or. It's not possible for a value of x to be both less than -30 and greater than 30.

 

[vela]

Not really. The best (only?) way is to consider two cases and see where each case takes you.

I think the OP is asking how you come up with the two cases in the first place. That is, how do you know it's supposed to be ##x < -\sqrt{a}## instead of ##x > -\sqrt{a}## other than simply memorizing?

The easiest way is using what @BvU has hinted at. Since ##\sqrt{x^2} = \lvert x \rvert##, after taking the square root of the original inequality, you have ##\lvert x \rvert > 30##. Then the definition of the absolute value leads to the two cases
$$ \lvert x \rvert > 30 \quad \Rightarrow \quad
\begin{cases}
x > 30, & x>0 \\
-x > 30, & x<0
\end{cases}$$ Negating the latter case gives you ##x < -30##.

 

[FactChecker]

I think the OP is asking how you come up with the two cases in the first place. That is, how do you know it's supposed to be ##x < -\sqrt{a}## instead of ##x > -\sqrt{a}## other than simply memorizing?

One way that works in many such inequalities is to try some cases in each section of the reals in question: x=0 and x=-1000 and see which ones work.

 

[mark2142]

The easiest way is using what @BvU has hinted at. Since x2=|x|, after taking the square root of the original inequality, you have |x|>30.

Thank you for clarifying but what is ##\sqrt 900##? Doesn't the inquality ##x^2 > 900## becomes ##|x| > \pm 30## after square rooting?

 

[Mark44]

Thank you for clarifying but what is ##\sqrt 900##? Doesn't the inquality ##x^2 > 900## becomes ##|x| > \pm 30## after square rooting?

##\sqrt{900} = 30##
If ##x^2 > 900## then ##|x| > 30##

The expression ##\sqrt{900}## represents the principal (i.e., positive) square root of 900. It's a common mistake that we see a lot where people think that the square root of a number is plus/minus some quantity. That is erroneous.

 

[mark2142]

##\sqrt{900} = 30##
If ##x^2 > 900## then ##|x| > 30##

The expression ##\sqrt{900}## represents the principal (i.e., positive) square root of 900. It's a common mistake that we see a lot where people think that the square root of a number is plus/minus some quantity. That is erroneous.1

https://math.stackexchange.com/a/547831/1109500
Is this what you mean? That we choose ##\sqrt x## to be a positive square root because it helps in solving inequalities and equations. And for that to be consistent all the way we want ##\sqrt {x^2}= |x| ## to be true.
Then it will be like @vela said. ##|x| >30## and by definition of absolute values we have x<-30 or x>+30.

 

[Mark44]

https://math.stackexchange.com/a/547831/1109500
Is this what you mean?

Yes

That we choose ##\sqrt x## to be a positive square root because it helps in solving inequalities and equations.

No, we choose ##\sqrt x ## to be the positive square root of x so that the square root operation will be a function. A function has only a single value. However, in more advanced mathematics the concept of multi-valued or vector-valued functions comes up.

And for that to be consistent all the way we want ##\sqrt {x^2}= |x| ## to be true.

I don't think that's the right explanation. By definition, |x| equals x if x is is nonnegative, and equals -x if x happens to be negative. With our convention, ##\sqrt{x^2}## has the same value as |x|.

Then it will be like @vela said. ##|x| >30## and by definition of absolute values we have x<-30 or x>+30.

 

[PeroK]

https://math.stackexchange.com/a/547831/1109500
Is this what you mean? That we choose ##\sqrt x## to be a positive square root because it helps in solving inequalities and equations. And for that to be consistent all the way we want ##\sqrt {x^2}= |x| ## to be true.
Then it will be like @vela said. ##|x| >30## and by definition of absolute values we have x<-30 or x>+30.

$$(-31)(-31) = (31)(31) > 900$$$$(-29)(-29) = (29)(29) < 900$$$$-31<-30<-29< 0 < 29 < 30 < 31$$

 

[mark2142]

don't think that's the right explanation. By definition, |x| equals x if x is is nonnegative, and equals -x if x happens to be negative. With our convention, x2 has the same value as |x|.

When you say ## \sqrt {x^2}= |x|## is this because ## \sqrt y## is taken as positive root where ##y=x^2## ? (And |x| is already positive. So both becomes equal).
Like ## \sqrt {(-2)^2}= \sqrt 4 = +2##
And ## \sqrt {2^2}= \sqrt 4 = +2##
So, ## \sqrt x^2 = |x|## for all x.

 

[PeroK]

So, ## \sqrt x^2 = |x|## for all x.

Exactly.

 

[Mark44]

When you say ## \sqrt {x^2}= |x|## is this because ## \sqrt y## is taken as positive root where ##y=x^2##?

It doesn't really have anything to do with ##y = x^2##.
If x < 0, then ##\sqrt{x^2} = \sqrt{(-x)^2} = -x##, which is a positive number.
If ##x \ge 0## then ##\sqrt{x^2} = x##, which is nonnegative.
Therefore, ##\sqrt{x^2}## produces the exact same result as |x|.

(And |x| is already positive. So both becomes equal).
Like ## \sqrt {(-2)^2}= \sqrt 4 = +2##
And ## \sqrt {2^2}= \sqrt 4 = +2##
So, ## \sqrt x^2 = |x|## for all x.

Yes, with this correction. You wrote ## \sqrt x^2## in the last line. I think I understand what you meant, but that wasn't consistent with the LaTeX that you wrote.
## \sqrt x^2## is rendered as if you had written ##(\sqrt x)^2##, which is defined only for ##x \ge 0## if the intention is for the real-valued square root.
The corrected version of that last line is ##\sqrt{x^2} = |x|##. In other words, you need braces around the x^2 part. So there is a difference between taking the square root first and then squaring the result, versus squaring x first, then taking the square root of that result.

 

[mark2142]

Exactly.

Ok. Great. And the fact that ## (-2)^{2*1/2}= -2## is true but we ignore it and say ## (-2)^{2*1/2}= |-2|=2##. Yes?

 

[PeroK]

Ok. Great. And the fact that ## (-2)^{2*1/2}= -2## is true but we ignore it and say ## (-2)^{2*1/2}= |-2|=2##. Yes?

No.

 

[Mark44]

Ok. Great. And the fact that ## (-2)^{2*1/2}= -2## is true but we ignore it and say ## (-2)^{2*1/2}= |-2|=2##. Yes?

No. The usual rules for exponents don't apply when you raise negative numbers to some power. The rule you apparently are using is ##(a^b)^c = a^{bc}## requires that a > 0.
For the expression you started with, the expressions ##((-2)^2)^{1/2}## and ##(-2)^{2 * 1/2}## produce different results, namely 2 and -2, respectively.

 

[PeroK]

Ok. Great. And the fact that ## (-2)^{2*1/2}= -2## is true but we ignore it and say ## (-2)^{2*1/2}= |-2|=2##. Yes?

Keep things simple:
$$(x^2)^{1/2} = \sqrt{x^2} = |x|$$$$x^{(2*\frac 1 2)} = x$$

 

[mark2142]

It doesn't really have anything to do with y=x2.

I meant to substitute ##x^2## with ##y## so to make it more clear. Square root of y is defined to be positive root. My explanation seems right.
I am not saying I don’t agree with yours. It’s just I get mine and it’s easy to remember.

 

[mark2142]

Thank you.