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Popular >

2(tan(x)+1)+3(1-tan(x))<-2(tan(x)-3)

Solution

2 (tan (x) + 1) + 3 (1 - tan (x)) < - 2 (tan (x) - 3)

Solution

- \frac{π}{2} + πn < x < \frac{π}{4} + πn

+2

Interval Notation

(- \frac{π}{2} + πn, \frac{π}{4} + πn)

Decimal Notation

- 1.57079 \dots + πn < x < 0.78539 \dots + πn

Solution steps

2 (tan (x) + 1) + 3 (1 - tan (x)) < - 2 (tan (x) - 3)

Let:

u = tan (x)

2 (u + 1) + 3 (1 - u) < - 2 (u - 3)

2 (u + 1) + 3 (1 - u) < - 2 (u - 3) : u < 1

2 (u + 1) + 3 (1 - u) < - 2 (u - 3)

Expand

2 (u + 1) + 3 (1 - u) : - u + 5

2 (u + 1) + 3 (1 - u)

Expand

2 (u + 1) : 2 u + 2

2 (u + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = u, c = 1

= 2 u + 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u + 2

= 2 u + 2 + 3 (1 - u)

Expand

3 (1 - u) : 3 - 3 u

3 (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = u

= 3 \cdot 1 - 3 u

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 u

= 2 u + 2 + 3 - 3 u

Simplify

2 u + 2 + 3 - 3 u : - u + 5

2 u + 2 + 3 - 3 u

Group like terms

= 2 u - 3 u + 2 + 3

Add similar elements:

2 u - 3 u = - u

= - u + 2 + 3

Add the numbers:

2 + 3 = 5

= - u + 5

= - u + 5

Expand

- 2 (u - 3) : - 2 u + 6

- 2 (u - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = u, c = 3

= - 2 u - (- 2) \cdot 3

Apply minus-plus rules

- (- a) = a

= - 2 u + 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= - 2 u + 6

- u + 5 < - 2 u + 6

Move

5

to the right side

- u + 5 < - 2 u + 6

Subtract

5

from both sides

- u + 5 - 5 < - 2 u + 6 - 5

Simplify

- u < - 2 u + 1

- u < - 2 u + 1

Move

2 u

to the left side

- u < - 2 u + 1

Add

2 u

to both sides

- u + 2 u < - 2 u + 1 + 2 u

Simplify

u < 1

u < 1

u < 1

Substitute back

u = tan (x)

tan (x) < 1

If

tan (x) < a

then

- \frac{π}{2} + πn < x < arctan (a) + πn

- \frac{π}{2} + πn < x < arctan (1) + πn

Simplify

arctan (1) : \frac{π}{4}

arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4}

- \frac{π}{2} + πn < x < \frac{π}{4} + πn

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