Algebra Examples

$\tan (x) > - 1$

Step 1

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x > \arctan (- 1)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x > - \frac{π}{4}$

Step 3

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 4

Simplify the expression to find the second solution.

Tap for more steps...

Step 4.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 4.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 5

Find the period of $\tan (x)$ .

Tap for more steps...

Step 5.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 5.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 5.4

Divide $π$ by $1$ .

$π$

Step 6

Add $π$ to every negative angle to get positive angles.

Tap for more steps...

Step 6.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 6.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 6.3

Combine fractions.

Tap for more steps...

Step 6.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 6.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 6.5

List the new angles.

$x = \frac{3 π}{4}$

Step 7

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 8

Consolidate the answers.

$x = \frac{3 π}{4} + π n$ , for any integer $n$

Step 9

Find the domain of $\tan (x) + 1$ .

Tap for more steps...

Step 9.1

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 9.2

The domain is all values of $x$ that make the expression defined.

${x | x \neq \frac{π}{2} + π n}$ $n$ , for any integer $n$

Step 10

Use each root to create test intervals.

$\frac{π}{2} < x < \frac{3 π}{4}$

$\frac{3 π}{4} < x < \frac{3 π}{2}$

$\frac{3 π}{2} < x < \frac{7 π}{4}$

Step 11

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 11.1

Test a value on the interval $\frac{π}{2} < x < \frac{3 π}{4}$ to see if it makes the inequality true.

Tap for more steps...

Step 11.1.1

Choose a value on the interval $\frac{π}{2} < x < \frac{3 π}{4}$ and see if this value makes the original inequality true.

$x = 2$

Step 11.1.2

Replace $x$ with $2$ in the original inequality.

$\tan (2) > - 1$

Step 11.1.3

The left side $- 2.18503986$ is not greater than the right side $- 1$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $\frac{3 π}{4} < x < \frac{3 π}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 11.2.1

Choose a value on the interval $\frac{3 π}{4} < x < \frac{3 π}{2}$ and see if this value makes the original inequality true.

$x = 4$

Step 11.2.2

Replace $x$ with $4$ in the original inequality.

$\tan (4) > - 1$

Step 11.2.3

The left side $1.15782128$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $\frac{3 π}{2} < x < \frac{7 π}{4}$ to see if it makes the inequality true.

Tap for more steps...

Step 11.3.1

Choose a value on the interval $\frac{3 π}{2} < x < \frac{7 π}{4}$ and see if this value makes the original inequality true.

$x = 5$

Step 11.3.2

Replace $x$ with $5$ in the original inequality.

$\tan (5) > - 1$

Step 11.3.3

The left side $- 3.380515$ is not greater than the right side $- 1$ , which means that the given statement is false.

False

Step 11.4

Compare the intervals to determine which ones satisfy the original inequality.

$\frac{π}{2} < x < \frac{3 π}{4}$ False

$\frac{3 π}{4} < x < \frac{3 π}{2}$ True

$\frac{3 π}{2} < x < \frac{7 π}{4}$ False

$\frac{π}{2} < x < \frac{3 π}{4}$ False

$\frac{3 π}{4} < x < \frac{3 π}{2}$ True

$\frac{3 π}{2} < x < \frac{7 π}{4}$ False

Step 12

The solution consists of all of the true intervals.

$\frac{3 π}{4} + π n < x < \frac{3 π}{2} + π n$ , for any integer $n$

Step 13