solution-set-of-log-x-2-x-x-x-0-

Question Number 178635 by infinityaction last updated on 19/Oct/22

solution set of \log_{x^{2}} (\frac{x}{∣ x ∣} - x) ⩾ 0

Commented by Frix last updated on 19/Oct/22

x ⩽ - 2 \lor 0 < x < 1

Commented by infinityaction last updated on 20/Oct/22

s i r s o l u t i o n

Answered by mr W last updated on 20/Oct/22

f o r x^{2} ⩾ 1 : \frac{x}{∣ x ∣} - x ⩾ 1

f o r x > 0 :

1 - x ⩾ 1 \Rightarrow x ⩽ 0 \Rightarrow c o n t r a d i t i o n!

f o r x < 0 :

- 1 - x ⩾ 1 \Rightarrow x ⩽ - 2 ✓

f o r x^{2} ⩽ 1 : 0 < \frac{x}{∣ x ∣} - x ⩽ 1

f o r x > 0 :

0 < 1 - x ⩽ 1 \Rightarrow x ⩾ 0 \Rightarrow 1 > x > 0 ✓

f o r x < 0 :

0 < - 1 - x ⩽ 1 \Rightarrow - 2 ⩽ x < - 1 \Rightarrow c o n t r a d i c t i o n

\Rightarrow s o l u t i o n : x ⩽ - 2 o r 0 < x < 1

Commented by infinityaction last updated on 21/Oct/22

t h a n k y o u s i r