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arcsin((sqrt(3))/2-(0.15)/x)>=-pi/2

Solution

arcsin (\frac{3}{2} - \frac{0.15}{x}) \geq - \frac{π}{2}

Solution

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

Interval Notation

(- \infty, - 1.11961 \dots] \cup [0.08038 \dots, \infty)

Solution steps

arcsin (\frac{3}{2} - \frac{0.15}{x}) \geq - \frac{π}{2}

arcsin (x) \geq a

then

x \geq sin (a)

\frac{3}{2} - \frac{0.15}{x} \geq sin (- \frac{π}{2})

sin (- \frac{π}{2}) = - 1

sin (- \frac{π}{2})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{2}) = - sin (\frac{π}{2})

= - sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= - 1

\frac{3}{2} - \frac{0.15}{x} \geq - 1

\frac{3}{2} - \frac{0.15}{x} \geq - 1 : x < 0 or x \geq 0.08038 \dots

\frac{3}{2} - \frac{0.15}{x} \geq - 1

Rewrite in standard form

\frac{3}{2} - \frac{0.15}{x} \geq - 1

Add

1

to both sides

\frac{3}{2} - \frac{0.15}{x} + 1 \geq - 1 + 1

Simplify

\frac{3}{2} - \frac{0.15}{x} + 1 \geq 0

Simplify

\frac{3}{2} - \frac{0.15}{x} + 1 : \frac{3 x - 0.3 + 2 x}{2 x}

\frac{3}{2} - \frac{0.15}{x} + 1

Convert element to fraction:

1 = \frac{1}{1}

= \frac{3}{2} - \frac{0.15}{x} + \frac{1}{1}

Least Common Multiplier of

2, x, 1 : 2 x

2, x, 1

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 1 : 2

2, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

1

Multiply each factor the greatest number of times it occurs in either

2

1

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 2 x

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 x

For

\frac{3}{2} :

multiply the denominator and numerator by

x

\frac{3}{2} = \frac{3 x}{2 x}

For

\frac{0.15}{x} :

multiply the denominator and numerator by

2

\frac{0.15}{x} = \frac{0.15 \cdot 2}{x \cdot 2} = \frac{0.3}{2 x}

For

\frac{1}{1} :

multiply the denominator and numerator by

2 x

\frac{1}{1} = \frac{1 \cdot 2 x}{1 \cdot 2 x} = \frac{2 x}{2 x}

= \frac{3 x}{2 x} - \frac{0.3}{2 x} + \frac{2 x}{2 x}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 x - 0.3 + 2 x}{2 x}

\frac{3 x - 0.3 + 2 x}{2 x} \geq 0

Multiply both sides by

2

\frac{2 ( 3 x - 0.3 + 2 x )}{2 x} \geq 0 \cdot 2

Simplify

\frac{3 x - 0.3 + 2 x}{x} \geq 0

\frac{3 x - 0.3 + 2 x}{x} \geq 0

Identify the intervals

Find the signs of the factors of

\frac{3 x - 0.3 + 2 x}{x}

Find the signs of

3 x - 0.3 + 2 x

3 x - 0.3 + 2 x = 0 : x = 0.08038 \dots

3 x - 0.3 + 2 x = 0

Move

0.3

to the right side

3 x - 0.3 + 2 x = 0

Add

0.3

to both sides

3 x - 0.3 + 2 x + 0.3 = 0 + 0.3

Simplify

3 x + 2 x = 0.3

3 x + 2 x = 0.3

Factor

3 x + 2 x : (3 + 2) x

3 x + 2 x

Factor out common term

x

= x (3 + 2)

(3 + 2) x = 0.3

Divide both sides by

3 + 2

(3 + 2) x = 0.3

Divide both sides by

3 + 2

\frac{( 3 + 2 ) x}{3 + 2} = \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} = \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} : x

\frac{( 3 + 2 ) x}{3 + 2}

Cancel the common factor:

3 + 2

= x

Simplify

\frac{0.3}{3 + 2} : 0.08038 \dots

\frac{0.3}{3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots + 2}

Add the numbers:

1.73205 \dots + 2 = 3.73205 \dots

= \frac{0.3}{3.73205 \dots}

Divide the numbers:

\frac{0.3}{3.73205 \dots} = 0.08038 \dots

= 0.08038 \dots

x = 0.08038 \dots

x = 0.08038 \dots

x = 0.08038 \dots

3 x - 0.3 + 2 x < 0 : x < 0.08038 \dots

3 x - 0.3 + 2 x < 0

Move

0.3

to the right side

3 x - 0.3 + 2 x < 0

Add

0.3

to both sides

3 x - 0.3 + 2 x + 0.3 < 0 + 0.3

Simplify

3 x + 2 x < 0.3

3 x + 2 x < 0.3

Factor

3 x + 2 x : (3 + 2) x

3 x + 2 x

Factor out common term

x

= x (3 + 2)

(3 + 2) x < 0.3

Divide both sides by

3 + 2

(3 + 2) x < 0.3

Divide both sides by

3 + 2

\frac{( 3 + 2 ) x}{3 + 2} < \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} < \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} : x

\frac{( 3 + 2 ) x}{3 + 2}

Cancel the common factor:

3 + 2

= x

Simplify

\frac{0.3}{3 + 2} : 0.08038 \dots

\frac{0.3}{3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots + 2}

Add the numbers:

1.73205 \dots + 2 = 3.73205 \dots

= \frac{0.3}{3.73205 \dots}

Divide the numbers:

\frac{0.3}{3.73205 \dots} = 0.08038 \dots

= 0.08038 \dots

x < 0.08038 \dots

x < 0.08038 \dots

x < 0.08038 \dots

3 x - 0.3 + 2 x > 0 : x > 0.08038 \dots

3 x - 0.3 + 2 x > 0

Move

0.3

to the right side

3 x - 0.3 + 2 x > 0

Add

0.3

to both sides

3 x - 0.3 + 2 x + 0.3 > 0 + 0.3

Simplify

3 x + 2 x > 0.3

3 x + 2 x > 0.3

Factor

3 x + 2 x : (3 + 2) x

3 x + 2 x

Factor out common term

x

= x (3 + 2)

(3 + 2) x > 0.3

Divide both sides by

3 + 2

(3 + 2) x > 0.3

Divide both sides by

3 + 2

\frac{( 3 + 2 ) x}{3 + 2} > \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} > \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} : x

\frac{( 3 + 2 ) x}{3 + 2}

Cancel the common factor:

3 + 2

= x

Simplify

\frac{0.3}{3 + 2} : 0.08038 \dots

\frac{0.3}{3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots + 2}

Add the numbers:

1.73205 \dots + 2 = 3.73205 \dots

= \frac{0.3}{3.73205 \dots}

Divide the numbers:

\frac{0.3}{3.73205 \dots} = 0.08038 \dots

= 0.08038 \dots

x > 0.08038 \dots

x > 0.08038 \dots

x > 0.08038 \dots

Find the signs of

x

x = 0

x < 0

x > 0

Find singularity points

Find the zeros of the denominator

x : x = 0

Summarize in a table:

3 x - 0.3 + 2 x x \frac{3 x - 0.3 + 2 x}{x} x < 0 - - + x = 0 - 0 Undefined 0 < x < 0.08038 \dots - + - x = 0.08038 \dots 0 + 0 x > 0.08038 \dots + + +

Identify the intervals that satisfy the required condition:

\geq 0

x < 0 or x = 0.08038 \dots or x > 0.08038 \dots

Merge Overlapping Intervals

x < 0 or x = 0.08038 \dots or x > 0.08038 \dots

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

x < 0

x = 0.08038 \dots

x < 0 or x = 0.08038 \dots

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

x < 0 or x = 0.08038 \dots

x > 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

Domain of

arcsin (\frac{3}{2} - \frac{0.15}{x}) : x \leq - 1.11961 \dots or x \geq 0.08038 \dots

Domain definition

Find known functions domain restrictions:

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

arcsin (f (x)) \Rightarrow - 1 \leq f (x) \leq 1

Solve

- 1 \leq (\frac{3}{2} - \frac{0.15}{x}) \leq 1 : x \leq - 1.11961 \dots or x \geq 0.08038 \dots

- 1 \leq (\frac{3}{2} - \frac{0.15}{x}) \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

- 1 \leq (\frac{3}{2} - \frac{0.15}{x}) and (\frac{3}{2} - \frac{0.15}{x}) \leq 1

- 1 \leq \frac{3}{2} - \frac{0.15}{x} : x < 0 or x \geq 0.08038 \dots

- 1 \leq \frac{3}{2} - \frac{0.15}{x}

Switch sides

\frac{3}{2} - \frac{0.15}{x} \geq - 1

Rewrite in standard form

\frac{3}{2} - \frac{0.15}{x} \geq - 1

Add

1

to both sides

\frac{3}{2} - \frac{0.15}{x} + 1 \geq - 1 + 1

Simplify

\frac{3}{2} - \frac{0.15}{x} + 1 \geq 0

Simplify

\frac{3}{2} - \frac{0.15}{x} + 1 : \frac{3 x - 0.3 + 2 x}{2 x}

\frac{3}{2} - \frac{0.15}{x} + 1

Convert element to fraction:

1 = \frac{1}{1}

= \frac{3}{2} - \frac{0.15}{x} + \frac{1}{1}

Least Common Multiplier of

2, x, 1 : 2 x

2, x, 1

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 1 : 2

2, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

1

Multiply each factor the greatest number of times it occurs in either

2

1

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 2 x

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 x

For

\frac{3}{2} :

multiply the denominator and numerator by

x

\frac{3}{2} = \frac{3 x}{2 x}

For

\frac{0.15}{x} :

multiply the denominator and numerator by

2

\frac{0.15}{x} = \frac{0.15 \cdot 2}{x \cdot 2} = \frac{0.3}{2 x}

For

\frac{1}{1} :

multiply the denominator and numerator by

2 x

\frac{1}{1} = \frac{1 \cdot 2 x}{1 \cdot 2 x} = \frac{2 x}{2 x}

= \frac{3 x}{2 x} - \frac{0.3}{2 x} + \frac{2 x}{2 x}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 x - 0.3 + 2 x}{2 x}

\frac{3 x - 0.3 + 2 x}{2 x} \geq 0

Multiply both sides by

2

\frac{2 ( 3 x - 0.3 + 2 x )}{2 x} \geq 0 \cdot 2

Simplify

\frac{3 x - 0.3 + 2 x}{x} \geq 0

\frac{3 x - 0.3 + 2 x}{x} \geq 0

Identify the intervals

Find the signs of the factors of

\frac{3 x - 0.3 + 2 x}{x}

Find the signs of

3 x - 0.3 + 2 x

3 x - 0.3 + 2 x = 0 : x = 0.08038 \dots

3 x - 0.3 + 2 x = 0

Move

0.3

to the right side

3 x - 0.3 + 2 x = 0

Add

0.3

to both sides

3 x - 0.3 + 2 x + 0.3 = 0 + 0.3

Simplify

3 x + 2 x = 0.3

3 x + 2 x = 0.3

Factor

3 x + 2 x : (3 + 2) x

3 x + 2 x

Factor out common term

x

= x (3 + 2)

(3 + 2) x = 0.3

Divide both sides by

3 + 2

(3 + 2) x = 0.3

Divide both sides by

3 + 2

\frac{( 3 + 2 ) x}{3 + 2} = \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} = \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} : x

\frac{( 3 + 2 ) x}{3 + 2}

Cancel the common factor:

3 + 2

= x

Simplify

\frac{0.3}{3 + 2} : 0.08038 \dots

\frac{0.3}{3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots + 2}

Add the numbers:

1.73205 \dots + 2 = 3.73205 \dots

= \frac{0.3}{3.73205 \dots}

Divide the numbers:

\frac{0.3}{3.73205 \dots} = 0.08038 \dots

= 0.08038 \dots

x = 0.08038 \dots

x = 0.08038 \dots

x = 0.08038 \dots

3 x - 0.3 + 2 x < 0 : x < 0.08038 \dots

3 x - 0.3 + 2 x < 0

Move

0.3

to the right side

3 x - 0.3 + 2 x < 0

Add

0.3

to both sides

3 x - 0.3 + 2 x + 0.3 < 0 + 0.3

Simplify

3 x + 2 x < 0.3

3 x + 2 x < 0.3

Factor

3 x + 2 x : (3 + 2) x

3 x + 2 x

Factor out common term

x

= x (3 + 2)

(3 + 2) x < 0.3

Divide both sides by

3 + 2

(3 + 2) x < 0.3

Divide both sides by

3 + 2

\frac{( 3 + 2 ) x}{3 + 2} < \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} < \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} : x

\frac{( 3 + 2 ) x}{3 + 2}

Cancel the common factor:

3 + 2

= x

Simplify

\frac{0.3}{3 + 2} : 0.08038 \dots

\frac{0.3}{3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots + 2}

Add the numbers:

1.73205 \dots + 2 = 3.73205 \dots

= \frac{0.3}{3.73205 \dots}

Divide the numbers:

\frac{0.3}{3.73205 \dots} = 0.08038 \dots

= 0.08038 \dots

x < 0.08038 \dots

x < 0.08038 \dots

x < 0.08038 \dots

3 x - 0.3 + 2 x > 0 : x > 0.08038 \dots

3 x - 0.3 + 2 x > 0

Move

0.3

to the right side

3 x - 0.3 + 2 x > 0

Add

0.3

to both sides

3 x - 0.3 + 2 x + 0.3 > 0 + 0.3

Simplify

3 x + 2 x > 0.3

3 x + 2 x > 0.3

Factor

3 x + 2 x : (3 + 2) x

3 x + 2 x

Factor out common term

x

= x (3 + 2)

(3 + 2) x > 0.3

Divide both sides by

3 + 2

(3 + 2) x > 0.3

Divide both sides by

3 + 2

\frac{( 3 + 2 ) x}{3 + 2} > \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} > \frac{0.3}{3 + 2}

Simplify

\frac{( 3 + 2 ) x}{3 + 2} : x

\frac{( 3 + 2 ) x}{3 + 2}

Cancel the common factor:

3 + 2

= x

Simplify

\frac{0.3}{3 + 2} : 0.08038 \dots

\frac{0.3}{3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots + 2}

Add the numbers:

1.73205 \dots + 2 = 3.73205 \dots

= \frac{0.3}{3.73205 \dots}

Divide the numbers:

\frac{0.3}{3.73205 \dots} = 0.08038 \dots

= 0.08038 \dots

x > 0.08038 \dots

x > 0.08038 \dots

x > 0.08038 \dots

Find the signs of

x

x = 0

x < 0

x > 0

Find singularity points

Find the zeros of the denominator

x : x = 0

Summarize in a table:

3 x - 0.3 + 2 x x \frac{3 x - 0.3 + 2 x}{x} x < 0 - - + x = 0 - 0 Undefined 0 < x < 0.08038 \dots - + - x = 0.08038 \dots 0 + 0 x > 0.08038 \dots + + +

Identify the intervals that satisfy the required condition:

\geq 0

x < 0 or x = 0.08038 \dots or x > 0.08038 \dots

Merge Overlapping Intervals

x < 0 or x = 0.08038 \dots or x > 0.08038 \dots

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

x < 0

x = 0.08038 \dots

x < 0 or x = 0.08038 \dots

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

x < 0 or x = 0.08038 \dots

x > 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

x < 0 or x \geq 0.08038 \dots

\frac{3}{2} - \frac{0.15}{x} \leq 1 : x \leq - 1.11961 \dots or x > 0

\frac{3}{2} - \frac{0.15}{x} \leq 1

Rewrite in standard form

\frac{3}{2} - \frac{0.15}{x} \leq 1

Subtract

1

from both sides

\frac{3}{2} - \frac{0.15}{x} - 1 \leq 1 - 1

Simplify

\frac{3}{2} - \frac{0.15}{x} - 1 \leq 0

Simplify

\frac{3}{2} - \frac{0.15}{x} - 1 : \frac{3 x - 0.3 - 2 x}{2 x}

\frac{3}{2} - \frac{0.15}{x} - 1

Convert element to fraction:

1 = \frac{1}{1}

= \frac{3}{2} - \frac{0.15}{x} - \frac{1}{1}

Least Common Multiplier of

2, x, 1 : 2 x

2, x, 1

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 1 : 2

2, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

1

Multiply each factor the greatest number of times it occurs in either

2

1

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 2 x

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 x

For

\frac{3}{2} :

multiply the denominator and numerator by

x

\frac{3}{2} = \frac{3 x}{2 x}

For

\frac{0.15}{x} :

multiply the denominator and numerator by

2

\frac{0.15}{x} = \frac{0.15 \cdot 2}{x \cdot 2} = \frac{0.3}{2 x}

For

\frac{1}{1} :

multiply the denominator and numerator by

2 x

\frac{1}{1} = \frac{1 \cdot 2 x}{1 \cdot 2 x} = \frac{2 x}{2 x}

= \frac{3 x}{2 x} - \frac{0.3}{2 x} - \frac{2 x}{2 x}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 x - 0.3 - 2 x}{2 x}

\frac{3 x - 0.3 - 2 x}{2 x} \leq 0

Multiply both sides by

2

\frac{2 ( 3 x - 0.3 - 2 x )}{2 x} \leq 0 \cdot 2

Simplify

\frac{3 x - 0.3 - 2 x}{x} \leq 0

\frac{3 x - 0.3 - 2 x}{x} \leq 0

Identify the intervals

Find the signs of the factors of

\frac{3 x - 0.3 - 2 x}{x}

Find the signs of

3 x - 0.3 - 2 x

3 x - 0.3 - 2 x = 0 : x = - 1.11961 \dots

3 x - 0.3 - 2 x = 0

Move

0.3

to the right side

3 x - 0.3 - 2 x = 0

Add

0.3

to both sides

3 x - 0.3 - 2 x + 0.3 = 0 + 0.3

Simplify

3 x - 2 x = 0.3

3 x - 2 x = 0.3

Factor

3 x - 2 x : (3 - 2) x

3 x - 2 x

Factor out common term

x

= x (3 - 2)

(3 - 2) x = 0.3

Divide both sides by

3 - 2

(3 - 2) x = 0.3

Divide both sides by

3 - 2

\frac{( 3 - 2 ) x}{3 - 2} = \frac{0.3}{3 - 2}

Simplify

\frac{( 3 - 2 ) x}{3 - 2} = \frac{0.3}{3 - 2}

Simplify

\frac{( 3 - 2 ) x}{3 - 2} : x

\frac{( 3 - 2 ) x}{3 - 2}

Cancel the common factor:

3 - 2

= x

Simplify

\frac{0.3}{3 - 2} : - 1.11961 \dots

\frac{0.3}{3 - 2}

Convert element to a decimal form

3 = 1.73205 \dots

= \frac{0.3}{1.73205 \dots - 2}

Subtract the numbers:

1.73205 \dots - 2 = - 0.26794 \dots

= \frac{0.3}{- 0.26794 \dots}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{0.3}{0.26794 \dots}

Divide the numbers:

\frac{0.3}{0.26794 \dots} = 1.11961 \dots

= - 1.11961 \dots

x = - 1.11961 \dots

x = - 1.11961 \dots

x = - 1.11961 \dots

3 x - 0.3 - 2 x < 0 : x > - 1.11961 \dots

3 x - 0.3 - 2 x < 0

Move

0.3

to the right side

3 x - 0.3 - 2 x < 0

Add

0.3

to both sides

3 x - 0.3 - 2 x + 0.3 < 0 + 0.3

Simplify

3 x - 2 x < 0.3

3 x - 2 x < 0.3

Factor

3 x - 2 x : (3 - 2) x

3 x - 2 x

Factor out common term

x

= x (3 - 2)

(3 - 2) x < 0.3

Multiply both sides by

- 1

(3 - 2) x < 0.3

Multiply both sides by -1 (reverse the inequality)

(3 - 2) x (- 1) > 0.3 (- 1)

Simplify

- (3 - 2) x > - 0.3

- (3 - 2) x > - 0.3

Divide both sides by

- 3 + 2

- (3 - 2) x > - 0.3

Divide both sides by

- 3 + 2

\frac{- ( 3 - 2 ) x}{- 3 + 2} > \frac{- 0.3}{- 3 + 2}

Simplify

\frac{- ( 3 - 2 ) x}{- 3 + 2} > \frac{- 0.3}{- 3 + 2}

Simplify

\frac{- ( 3 - 2 ) x}{- 3 + 2} : x

\frac{- ( 3 - 2 ) x}{- 3 + 2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{( 3 - 2 ) x}{- 3 + 2}

2 - 3 = - (3 - 2)

= \frac{( 3 - 2 ) x}{- ( 3 - 2 )}

Refine

= - \frac{( 3 - 2 ) x}{3 - 2}

Cancel the common factor:

3 - 2

= - (- x)

Apply rule

- (- a) = a

= x

Simplify

\frac{- 0.3}{- 3 + 2} : - 1.11961 \dots

\frac{- 0.3}{- 3 + 2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{0.3}{- 3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= - \frac{0.3}{2 - 1.73205 \dots}

Add/Subtract the numbers:

- 1.73205 \dots + 2 = 0.26794 \dots

= - \frac{0.3}{0.26794 \dots}

Divide the numbers:

\frac{0.3}{0.26794 \dots} = 1.11961 \dots

= - 1.11961 \dots

x > - 1.11961 \dots

x > - 1.11961 \dots

x > - 1.11961 \dots

3 x - 0.3 - 2 x > 0 : x < - 1.11961 \dots

3 x - 0.3 - 2 x > 0

Move

0.3

to the right side

3 x - 0.3 - 2 x > 0

Add

0.3

to both sides

3 x - 0.3 - 2 x + 0.3 > 0 + 0.3

Simplify

3 x - 2 x > 0.3

3 x - 2 x > 0.3

Factor

3 x - 2 x : (3 - 2) x

3 x - 2 x

Factor out common term

x

= x (3 - 2)

(3 - 2) x > 0.3

Multiply both sides by

- 1

(3 - 2) x > 0.3

Multiply both sides by -1 (reverse the inequality)

(3 - 2) x (- 1) < 0.3 (- 1)

Simplify

- (3 - 2) x < - 0.3

- (3 - 2) x < - 0.3

Divide both sides by

- 3 + 2

- (3 - 2) x < - 0.3

Divide both sides by

- 3 + 2

\frac{- ( 3 - 2 ) x}{- 3 + 2} < \frac{- 0.3}{- 3 + 2}

Simplify

\frac{- ( 3 - 2 ) x}{- 3 + 2} < \frac{- 0.3}{- 3 + 2}

Simplify

\frac{- ( 3 - 2 ) x}{- 3 + 2} : x

\frac{- ( 3 - 2 ) x}{- 3 + 2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{( 3 - 2 ) x}{- 3 + 2}

2 - 3 = - (3 - 2)

= \frac{( 3 - 2 ) x}{- ( 3 - 2 )}

Refine

= - \frac{( 3 - 2 ) x}{3 - 2}

Cancel the common factor:

3 - 2

= - (- x)

Apply rule

- (- a) = a

= x

Simplify

\frac{- 0.3}{- 3 + 2} : - 1.11961 \dots

\frac{- 0.3}{- 3 + 2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{0.3}{- 3 + 2}

Convert element to a decimal form

3 = 1.73205 \dots

= - \frac{0.3}{2 - 1.73205 \dots}

Add/Subtract the numbers:

- 1.73205 \dots + 2 = 0.26794 \dots

= - \frac{0.3}{0.26794 \dots}

Divide the numbers:

\frac{0.3}{0.26794 \dots} = 1.11961 \dots

= - 1.11961 \dots

x < - 1.11961 \dots

x < - 1.11961 \dots

x < - 1.11961 \dots

Find the signs of

x

x = 0

x < 0

x > 0

Find singularity points

Find the zeros of the denominator

x : x = 0

Summarize in a table:

3 x - 0.3 - 2 x x \frac{3 x - 0.3 - 2 x}{x} x < - 1.11961 \dots + - - x = - 1.11961 \dots 0 - 0 - 1.11961 \dots < x < 0 - - + x = 0 - 0 Undefined x > 0 - + -

Identify the intervals that satisfy the required condition:

\leq 0

x < - 1.11961 \dots or x = - 1.11961 \dots or x > 0

Merge Overlapping Intervals

x \leq - 1.11961 \dots or x > 0

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

x < - 1.11961 \dots

x = - 1.11961 \dots

x \leq - 1.11961 \dots

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

x \leq - 1.11961 \dots

x > 0

x \leq - 1.11961 \dots or x > 0

x \leq - 1.11961 \dots or x > 0

x \leq - 1.11961 \dots or x > 0

Combine the intervals

(x < 0 or x \geq 0.08038 \dots) and (x \leq - 1.11961 \dots or x > 0)

Merge Overlapping Intervals

x < 0 or x \geq 0.08038 \dots and x \leq - 1.11961 \dots or x > 0

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

x < 0 or x \geq 0.08038 \dots

and

x \leq - 1.11961 \dots or x > 0

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

Find undefined (singularity) points:

x = 0

arcsin (\frac{3}{2} - \frac{0.15}{x})

Take the denominator(s) of

arcsin (\frac{3}{2} - \frac{0.15}{x})

and compare to zero

x = 0

The following points are undefined

x = 0

Combine real regions and undefined points for final function domain

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

Combine the intervals

x < 0 or x \geq 0.08038 \dots and x \leq - 1.11961 \dots or x \geq 0.08038 \dots

Merge Overlapping Intervals

x < 0 or x \geq 0.08038 \dots and x \leq - 1.11961 \dots or x \geq 0.08038 \dots

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

x < 0 or x \geq 0.08038 \dots

and

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

x \leq - 1.11961 \dots or x \geq 0.08038 \dots

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