Question

When you square root an inequality, such as this one:
\[x^2 \le 4\]
Do you flip the inequality sign? How would I go about solving this?

Answer 1

\[\left| x \right|\leq 2\]
\[-2 \leq x \leq 2\]

Answer 2

\[x^2 \leq a^2\]
\[\left| x \right| \leq a,-a \leq x \leq a\]
\[x^2 \geq a^2,x \leq -a,x \geq a\]

Answer 3

How come I never learned about this??? So is this a rule that you have to memorize, because logically, it isn't making sense to me

Answer 4

i tell you another way.
\[x^2 \leq 4,-2 \leq x \leq 2\]
or x lies between -2 and 2
take any number between -2 and 2
say-1 ,0 ,1
\[\left( -1 \right)^2=1 \leq 4\]
\[0^2=0 \leq 4\]
\[1^2=1 \leq4\]
now i take any number beyond this interval
say -3
\[\left( -3 \right)^2=9 \geq 4\]
similarly \[3^2=9 \geq 4\]
now it should be clear to you.

Answer 5

i have given you general formula.

Answer 6

Naturally it might be tempting to say
\[\Large \sqrt{x^2} = x\]
which is not true for all real numbers. It's only true if x is not negative. To ensure the output is nonnegative, we use absolute value brackets


So it's really
\[\Large \sqrt{x^2} = |x|\]
which is true for all real values x (even when x is negative)

Answer 7

So that's why after square rooting both sides, we go from
\[\Large x^2 < 4\]
to
\[\Large |x| < 2\]

Answer 8

Then you'll use these rules

http://www.mhhe.com/math/devmath/dugopolski/inter/instructor/ppt/ppt02/img017.jpg

to break things down further

Answer 9

Just to show some concrete examples of when this is true for integers (also true for nonintegers like 1/2 too):
\[x^2 \le 4\]
I'll go ahead and plug in all of the integers:
\[(-2)^2\le 4\]\[(-1)^2 \le 4\]\[0^2 \le 4\]\[1^2\le 4\]\[2^2\le 4\]
we can see since the square root makes negative numbers positive, it sorta "bounces" or "folds" the boundary of the interval [-2,2] back onto itself [0,2]

So when you take the square root of all the values to keep the interval the same since\(\sqrt{4}=2\)and \(\sqrt{4}=-2\), we have:

\[-2 \le -2 \le 2\]\[-2 \le -1 \le 2\]\[-2 \le 0 \le 2\]\[-2 \le 1 \le 2\]\[-2 \le 2 \le 2\]

This might confuse you at first, but I think it should help make it make sense for you @steve816 if you give it a little time and/or ask me questions I'll be happy to help.

Answer 10

Thanks for the response @kainui, it is much more clear now.
I have no idea why I didn't know about this until I started calculus, and seems like most teachers never go over this case during inequality unit in algebra.