1. ## Absolute Value Inequality

|x^2-6x-1|<6

What I did was:

-6<x^2-6x-1<6

-6<x^2-6x-1
-5<x^2-6x
-x^2+6x-5<0
(x-5)(x-1)
x=5, x=1
1<x<5

x^2-6x-1<6
x^2-6x-7<0
(x-7)(x+1)<0

-1<x<7

The answer given is -1<x<1 or 5<x<7

Do I have to sketch a graph to visualize it or something to understand it better?

2. Originally Posted by fuzzy
|x^2-6x-1|<6

What I did was:

-6<x^2-6x-1<6

-6<x^2-6x-1
-5<x^2-6x
-x^2+6x-5<0
(x-5)(x-1)
x=5, x=1
1<x<5
all was fine but this last line is wrong. Think why it should be $x\in(-\infty,1)\cup(5,\infty)$ and not 1<x<5.

x^2-6x-1<6
x^2-6x-7<0
(x-7)(x+1)<0

-1<x<7

The answer given is -1<x<1 or 5<x<7

Do I have to sketch a graph to visualize it or something to understand it better?
did this help?

3. Originally Posted by fuzzy
Do I have to sketch a graph to visualize it or something to understand it better?
You could do that, graph $\displaystyle y = |x^2-6x-1|$

Then draw a line through y=6, discard anything above this line.

4. Originally Posted by abhishekkgp
did this help?
Okay. Is the second solution correct?

5. Originally Posted by fuzzy
Okay. Is the second solution correct?
yup!