find-the-solution-of-4-x-2-x-x-3-4x-

Question Number 80433 by jagoll last updated on 03/Feb/20

f i n d t h e s o l u t i o n o f

\sqrt{4 - x} - 2 ⩽ x ∣ x - 3 ∣ + 4 x

Commented by john santu last updated on 03/Feb/20

(1) 4 - x ⩾ 0 \Rightarrow x ⩽ 4

(2) \sqrt{4 - x} ⩽ x ∣ x - 3 ∣ + 4 x + 2

f o r x ⩾ 3 \Rightarrow \sqrt{4 - x} ⩽ x (x - 3 + 4) + 2

\sqrt{4 - x} ⩽ x (x + 1) + 2

Commented by john santu last updated on 03/Feb/20

s o l u t i o n 0 ⩽ x ⩽ 4

Commented by jagoll last updated on 03/Feb/20

p l e a s e e x p l a i n m e

Commented by jagoll last updated on 03/Feb/20

m i s t e r W p l e a s e h e l p m e t h e

q u e s t i o n

Commented by mr W last updated on 03/Feb/20

s o l u t i o n f r o m s a n t u s i r 0 ⩽ x ⩽ 4 i s c o r r e c t .

Commented by jagoll last updated on 03/Feb/20

p l e a s e e x p l a i n m e . h o w g e t i t ?

Answered by mr W last updated on 03/Feb/20

x ⩽ 4

i f 3 ⩽ x ⩽ 4 :

\sqrt{4 - x} - 2 ⩽ x (x - 3) + 4 x

\sqrt{4 - x} ⩽ x^{2} + x + 2

L H S ⩽ 1

R H S ⩾ 3^{2} + 3 + 2 = 14

\Rightarrow L H S ⩽ R H S i s t r u e

i f x < 3 :

\sqrt{4 - x} - 2 ⩽ x (3 - x) + 4 x

\sqrt{4 - x} - 2 ⩽ x (7 - x)

L H S ⩽ 0 w h e n x ⩾ 0

L H S > 0 w h e n x < 0

R H S ⩾ 0 w h e n x ⩾ 0 (x < 3)

R H S < 0 w h e n x < 0

w h e n x ⩾ 0 :

L H S ⩽ 0 a n d R H S ⩾ 0 \Rightarrow L H S ⩽ R H S i s t r u e .

w h e n x < 0 :

L H S > 0 a n d R H S < 0 \Rightarrow L H S ⩽ R H S i s n o t t r u e .

t h a t m e a n s t h e i n e q u a l i t y i s v a l i d, w h e n

3 ⩽ x ⩽ 4 o r 0 ⩽ x < 3

\Rightarrow 0 ⩽ x ⩽ 4

Commented by mr W last updated on 03/Feb/20

b y h a n d l i n g i n e q u a l i t y {i t}^{'} s n o t g o o d

i d e a t o s q u a r e b o t h s i d e s, b e c a u s e t h a t

w i l l c h a n g e t h e r a n g e o f v a l i d i t y o f

t h e i n e q u a l i t y!

Commented by john santu last updated on 03/Feb/20

t h a n k y o u s i r

Commented by jagoll last updated on 03/Feb/20

t h a n k y o u m i s t e r

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