simple math question

[joonkimdds]

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Hey guys, I learned algebra 1 about 10 yrs ago and forgot everything.

I remember that there was a x axis and sometimes we say x is either less than -3 or greater than 3 and then we color everything on the left side of -3 and right side of 3.

and also, sometimes we say x is between -3 and 3, and color everything in between them.

I forgot what equations can cause them. I am sure it was very simple algebra equation but I just can't remember them.


And when we say square root of 16, is it just +4 or both + and - 4?

I had one question that says it's only + but then this another question i have says x square < 16 then -4<x<4. I don't know how they derived this. And even if square root of 16 is both + and -, how do we know that the symbol < changes to > when it comes to -4?

Thanks.

 

[Shinpe]

|x|<10 or x^2<100 both mean -10<x<10
|x|>10 or x^2>100 both mean x<-10 or x>10
sqrt(any (+) number(you can't have (-) inside)) = positive
x^2=100 would give you x= +/- sqrt(100), which gives you +/- 10

So in those two problems, one of them was probably just x=sqrt(16) while the other one you're starting with x^2 so when you take the sqrt, you get +/-

 

[Streetwolf]

^Correct but to clarify, it's because sqrt(x) is only a function if you disregard the negative roots. Sqrt(16) is definitely 4 but what about -4? Well (-4)^2 = 16 so yeah that looks good. The only problem is that we want sqrt(x) to be a function. How is something a function? Remember your vertical line test. You can only have (at most) one y value for any given x value.

Sqrt(x) fails this if you let the negative roots work. Sqrt(16) has x = 16. What is y? Well y can be 4 or -4. But for this to be a function we can only allow ONE y value, so we make it the positive one. If you want the negative square root of 16 then you say -sqrt(16).

Thus when you have a problem that says x^2 = 16 you can have -4 and 4... because you are not using the square root function. If you plug 4 or -4 into this equation THEY BOTH WORK.

If you have sqrt(16) = x... well then we only want +4.

And you switch signs when you consider the negative:

x^2 > 16
Keep the inequality with the positive answer:
x > 4
Flip the inequality with the negative answer:
x < -4

Again...

x^2 < 9
Keep the inequality with the positive answer:
x < 3
Flip the inequality with the negative answer:
x > -3

 

[FutureInternist]

I had one question that says it's only + but then this another question i have says x square < 16 then -4<x<4. I don't know how they derived this. And even if square root of 16 is both + and -, how do we know that the symbol < changes to > when it comes to -4?

You can also use "test points". With -4 & +4 as your "critical" points, you pick test points to see what sign to use. (with one point less than the smallest critical, one point bigger than the largest critical & one in b/w the criticals)

So for the ineq x^2 < 16 (with critical points -4 & +4)
Use the test points -5, 0 & 6

• -5^2 = 25 which is greater than 16 so all numbers less than -4 are excluded
• 0^2 = 0, less than 16 so all points in b/w the critical points are allowed
• 6^2 = 36, greater than 16 so all points greater than +4 are excluded.
  

Also keep in mind, when shading, that if the equation is just < or > then you use dashed lines, while if it is an < or > (with the equal sign) then it is a solid line.