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Popular >

-1<(cos(x))/7 <1

Solution

- 1 < \frac{cos ( x )}{7} < 1

Solution

True for all x \in R

Interval Notation

(- \infty, \infty)

Solution steps

- 1 < \frac{cos ( x )}{7} < 1

a < u < b

then

a < u and u < b

- 1 < \frac{cos ( x )}{7} and \frac{cos ( x )}{7} < 1

- 1 < \frac{cos ( x )}{7} :

True for all

x \in R

- 1 < \frac{cos ( x )}{7}

Switch sides

\frac{cos ( x )}{7} > - 1

Multiply both sides by

7

\frac{cos ( x )}{7} > - 1

Multiply both sides by

7

\frac{7 cos ( x )}{7} > 7 (- 1)

Simplify

cos (x) > - 7

cos (x) > - 7

Range of

cos (x) : - 1 \leq cos (x) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) > - 7 and - 1 \leq cos (x) \leq 1 : - 1 \leq cos (x) \leq 1

Let

y = cos (x)

Combine the intervals

y > - 7 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y > - 7 and - 1 \leq y \leq 1

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

y > - 7

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

\frac{cos ( x )}{7} < 1 :

True for all

x \in R

\frac{cos ( x )}{7} < 1

Multiply both sides by

7

\frac{cos ( x )}{7} < 1

Multiply both sides by

7

\frac{7 cos ( x )}{7} < 1 \cdot 7

Simplify

cos (x) < 7

cos (x) < 7

Range of

cos (x) : - 1 \leq cos (x) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) < 7 and - 1 \leq cos (x) \leq 1 : - 1 \leq cos (x) \leq 1

Let

y = cos (x)

Combine the intervals

y < 7 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < 7 and - 1 \leq y \leq 1

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

y < 7

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

True for all x \in R and True for all x \in R

Merge Overlapping Intervals

True for all x \in R and True for all x \in R

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

True for all

x \in R

and

True for all

x \in R

True for all x \in R

True for all x \in R

Popular Examples

sec(θ)=(sqrt(5))/2 \land sin(θ)<= 0

sec (θ) = \frac{5}{2} and sin (θ) \leq 0

tan(θ)<0\land sin(θ)>0

tan (θ) < 0 and sin (θ) > 0

cos(x)-sin(x)>0\land cos(x)>0

cos (x) - sin (x) > 0 and cos (x) > 0

sin(θ)<0\land sec(θ)<0

sin (θ) < 0 and sec (θ) < 0

0<= 1-cos(θ)<= 0.8

0 \leq 1 - cos (θ) \leq 0.8