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Popular >

cos(x^2)0<x<sqrt(x)

Solution

cos (x^{2}) 0 < x < x

Solution

0 < x < 1

Interval Notation

(0, 1)

Solution steps

cos (x^{2}) \cdot 0 < x < x

a < u < b

then

a < u and u < b

cos (x^{2}) \cdot 0 < x and x < x

cos (x^{2}) \cdot 0 < x : x > 0

cos (x^{2}) \cdot 0 < x

Apply rule

0 \cdot a = 0

0 < x

Move

0

to the right side

0 < x

Simplify

< x

< x

Move

x

to the left side

< x

Subtract

x

from both sides

- x < x - x

Simplify

- x < 0

- x < 0

Multiply both sides by

- 1

- x < 0

Multiply both sides by -1 (reverse the inequality)

(- x) (- 1) > 0 \cdot (- 1)

Simplify

x > 0

x > 0

x < x : 0 < x < 1

x < x

Switch sides

x > x

f (x) > g (x)

then

g (x) < 0 or g (x) \geq 0

For

x < 0 : x < 0

x < 0 and x > x

x > x :

True for all

x

x > x

If n is even,

for all

u

x > 0

x > 0 and 0 > x

then

x > x

True for all x

Combine the intervals

x < 0 and True for all x

Merge Overlapping Intervals

x < 0 and True for all x

The intersection of two intervals is the set of numbers which are in both intervals

x < 0

and

True for all

x

x < 0

x < 0

For

x \geq 0 : 0 < x < 1

x \geq 0 and x > x

x \geq 0

x > x : 0 < x < 1

Square both sides

(x)^{2} > x^{2}

Simplify

x > x^{2}

x > x^{2} : 0 < x < 1

x > x^{2}

Rewrite in standard form

x > x^{2}

Subtract

x^{2}

from both sides

x - x^{2} > x^{2} - x^{2}

Simplify

x - x^{2} > 0

x - x^{2} > 0

Factor

x - x^{2} : - x (x - 1)

x - x^{2}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

x^{2} = xx

= - xx + x

Factor out common term

- x

= - x (x - 1)

- x (x - 1) > 0

Multiply both sides by

- 1

(reverse the inequality)

(- x (x - 1)) (- 1) < 0 \cdot (- 1)

Simplify

x (x - 1) < 0

Identify the intervals

Find the signs of the factors of

x (x - 1)

Find the signs of

x

x = 0

x < 0

x > 0

Find the signs of

x - 1

x - 1 = 0 : x = 1

x - 1 = 0

Move

1

to the right side

x - 1 = 0

Add

1

to both sides

x - 1 + 1 = 0 + 1

Simplify

x = 1

x = 1

x - 1 < 0 : x < 1

x - 1 < 0

Move

1

to the right side

x - 1 < 0

Add

1

to both sides

x - 1 + 1 < 0 + 1

Simplify

x < 1

x < 1

x - 1 > 0 : x > 1

x - 1 > 0

Move

1

to the right side

x - 1 > 0

Add

1

to both sides

x - 1 + 1 > 0 + 1

Simplify

x > 1

x > 1

Summarize in a table:

x x - 1 x (x - 1) x < 0 - - + x = 0 0 - 0 0 < x < 1 + - - x = 1 + 00 x > 1 + + +

Identify the intervals that satisfy the required condition:

< 0

0 < x < 1

0 < x < 1

0 < x < 1

Combine the intervals

x \geq 0 and 0 < x < 1

Merge Overlapping Intervals

0 < x < 1 and x \geq 0

The intersection of two intervals is the set of numbers which are in both intervals

0 < x < 1

and

x \geq 0

0 < x < 1

0 < x < 1

Find singularity points

Find non-negative values for radicals:

x \geq 0

f (x) \Rightarrow f (x) \geq 0

For

x : x \geq 0

x \geq 0

Combine the intervals

(0 < x < 1 or x < 0) and x \geq 0

Merge Overlapping Intervals

x < 0 or 0 < x < 1 and x \geq 0

The union of two intervals is the set of numbers which are in either interval

0 < x < 1

x < 0

x < 0 or 0 < x < 1

The intersection of two intervals is the set of numbers which are in both intervals

x < 0 or 0 < x < 1

and

x \geq 0

0 < x < 1

0 < x < 1

Verify Solutions:

0 < x < 1

True

Check the solutions by plugging them into

x < x

Remove the ones that don't agree with the equation.

Verify

x < x

for

0 < x < 1 :

True

Plug in

x = 0.5 :

True

0.5 < 0.5

0.5 = 0.70710 \dots

0.5

0.5 = 0.70710 \dots

= 0.70710 \dots

0.5 < 0.70710 \dots

True

0 < x < 1

The solution is

0 < x < 1

Combine the intervals

x > 0 and 0 < x < 1

Merge Overlapping Intervals

x > 0 and 0 < x < 1

The intersection of two intervals is the set of numbers which are in both intervals

x > 0

and

0 < x < 1

0 < x < 1

0 < x < 1

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cos(θ)= 3/7 \land sin(θ)>0

cos (θ) = \frac{3}{7} and sin (θ) > 0

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cosh (θ) = \frac{26}{7} and θ < 0, sinh (θ)

-pi/2 <arcsin(x)< pi/2

- \frac{π}{2} < arcsin (x) < \frac{π}{2}

-sqrt(2)<= cos(2x)<= sqrt(2)

- 2 \leq cos (2 x) \leq 2