how-many-integer-number-satisfy-the-equation-sin-x-x-2-x-2-4x-5-0-

Question Number 100130 by bemath last updated on 25/Jun/20

how many integer number

satisfy the equation

\frac{\sin x - ∣ x + 2 ∣}{x^{2} - 4 x - 5} ⩾ 0

Answered by mr W last updated on 25/Jun/20

l e t f (x) = \sin x - ∣ x + 2 ∣

f o r x ⩽ - 2 :

f (x) = \sin x + x + 2

f^{'} (x) = \cos x + 1 ⩾ 0 \Rightarrow i n c r e a s i n g

f (x) ⩽ f (- 2) = \sin (- 2) \approx - 0.909

f o r x ⩾ - 2 :

f (x) = \sin x - x - 2

f^{'} (x) = \cos x - 1 ⩽ 0 \Rightarrow d e c r e a s i n g

f (x) ⩽ f (- 2) = \sin (- 2) \approx - 0.909

\Rightarrow f (x) = \sin x - ∣ x + 2 ∣⩽ \sin (- 2) \approx - 0.909 < 0

\frac{\sin x - ∣ x + 2 ∣}{x^{2} - 4 x - 5} ⩾ 0

s i n c e a l w a y s \sin x - ∣ x + 2 ∣< 0

\Rightarrow x^{2} - 4 x - 5 < 0

i . e . (x + 1) (x - 5) < 0

\Rightarrow - 1 < x < 5

\Rightarrow x = 0, 1, 2, 3, 4 \Rightarrow 5 i n t e g e r s

Commented by Rasheed.Sindhi last updated on 25/Jun/20

T_{H_{A} N} k_{S}

S_{I} R

!

Commented by bobhans last updated on 26/Jun/20

cooll great

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