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90-arctan((3L)/4)>= 40

Solution

90 - arctan (\frac{3 L}{4}) \geq 40

True for all L \in R

Interval Notation

(- \infty, \infty)

Solution steps

90 - arctan (\frac{3 L}{4}) \geq 40

Move

90

to the right side

90 - arctan (\frac{3 L}{4}) \geq 40

Subtract

90

from both sides

90 - arctan (\frac{3 L}{4}) - 90 \geq 40 - 90

Simplify

- arctan (\frac{3 L}{4}) \geq - 50

- arctan (\frac{3 L}{4}) \geq - 50

Multiply both sides by

- 1

- arctan (\frac{3 L}{4}) \geq - 50

Multiply both sides by -1 (reverse the inequality)

(- arctan (\frac{3 L}{4})) (- 1) \leq (- 50) (- 1)

Simplify

arctan (\frac{3 L}{4}) \leq 50

arctan (\frac{3 L}{4}) \leq 50

Range of

arctan (\frac{3 L}{4}) : - \frac{π}{2} < arctan (\frac{3 L}{4}) < \frac{π}{2}

Function range definition

The range of the basic

arctan

function is

- \frac{π}{2} < arctan (\frac{3 L}{4}) < \frac{π}{2}

- \frac{π}{2} < arctan (\frac{3 L}{4}) < \frac{π}{2}

arctan (\frac{3 L}{4}) \leq 50 and - \frac{π}{2} < arctan (\frac{3 L}{4}) < \frac{π}{2} : - \frac{π}{2} < arctan (\frac{3 L}{4}) < \frac{π}{2}

Let

y = arctan (\frac{3 L}{4})

Combine the intervals

y \leq 50 and - \frac{π}{2} < y < \frac{π}{2}

Merge Overlapping Intervals

y \leq 50 and - \frac{π}{2} < y < \frac{π}{2}

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

y \leq 50

and

- \frac{π}{2} < y < \frac{π}{2}

- \frac{π}{2} < y < \frac{π}{2}

- \frac{π}{2} < y < \frac{π}{2}

True for all L

True for all L \in R