Interval Notation with Absolute Value?

Question

Write in Interval notation:


3 absolute value(4-5x) less than equal to 9

and


1/x-x > 0


and


(x+2)^2 <25

Answer 1

3*|4-5x|<= 9

two problems === 3*(4-5x)<=9 and 3* (-(4-5x))<=9

3 |4-5x| <= 9

3 * (4-5x) <=9
12 - 15x <=9
-15x <= -3
x >= 1/5

3* (-(4-5x))<=9
3*(-4+5x)<=9
-12 + 15x <= 9
15x <= 21
x <= 7/5

so 1/5<=x<=7/5

in interval notation that is [1/5,7/5]

1/x -x >0 multiply everything by x
x/x - x^2 >0
1 - x^2 > 0
(1-x)(1+x) >0
1-x >0
-x > -1
x < 1

1+x >0
x > -1

in interval notation it is (-1,1)