Trigonometry Examples

$- 2 x^{2} + 3 x + 5 > 0$

Step 1

Convert the inequality to an equation.

$- 2 x^{2} + 3 x + 5 = 0$

Step 2

Factor the left side of the equation.

Tap for more steps...

Step 2.1

Factor $- 1$ out of $- 2 x^{2} + 3 x + 5$ .

Tap for more steps...

Step 2.1.1

Factor $- 1$ out of $- 2 x^{2}$ .

$- (2 x^{2}) + 3 x + 5 = 0$

Step 2.1.2

Factor $- 1$ out of $3 x$ .

$- (2 x^{2}) - (- 3 x) + 5 = 0$

Step 2.1.3

Rewrite $5$ as $- 1 (- 5)$ .

$- (2 x^{2}) - (- 3 x) - 1 \cdot - 5 = 0$

Step 2.1.4

Factor $- 1$ out of $- (2 x^{2}) - (- 3 x)$ .

$- (2 x^{2} - 3 x) - 1 \cdot - 5 = 0$

Step 2.1.5

Factor $- 1$ out of $- (2 x^{2} - 3 x) - 1 (- 5)$ .

$- (2 x^{2} - 3 x - 5) = 0$

Step 2.2

Factor.

Tap for more steps...

Step 2.2.1

Factor by grouping.

Tap for more steps...

Step 2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = - 3$ .

Tap for more steps...

Step 2.2.1.1.1

Factor $- 3$ out of $- 3 x$ .

$- (2 x^{2} - 3 x - 5) = 0$

Step 2.2.1.1.2

Rewrite $- 3$ as $2$ plus $- 5$

$- (2 x^{2} + (2 - 5) x - 5) = 0$

Step 2.2.1.1.3

Apply the distributive property.

$- (2 x^{2} + 2 x - 5 x - 5) = 0$

Step 2.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 x^{2} + 2 x) - 5 x - 5) = 0$

Step 2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (2 x (x + 1) - 5 (x + 1)) = 0$

Step 2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$- ((x + 1) (2 x - 5)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$- (x + 1) (2 x - 5) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$2 x - 5 = 0$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $2 x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 5.2

Solve $2 x - 5 = 0$ for $x$ .

Tap for more steps...

Step 5.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 5.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 6

The final solution is all the values that make $- (x + 1) (2 x - 5) = 0$ true.

$x = - 1, \frac{5}{2}$

Step 7

Use each root to create test intervals.

$x < - 1$

$- 1 < x < \frac{5}{2}$

$x > \frac{5}{2}$

Step 8

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 8.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Tap for more steps...

Step 8.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 8.1.2

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

$- 2 {(- 4)}^{2} + 3 (- 4) + 5 > 0$

Step 8.1.3

The left side $- 39$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $- 1 < x < \frac{5}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 8.2.1

Choose a value on the interval $- 1 < x < \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 8.2.2

Replace $x$ with $0$ in the original inequality.

$- 2 {(0)}^{2} + 3 (0) + 5 > 0$

Step 8.2.3

The left side $5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $x > \frac{5}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 8.3.1

Choose a value on the interval $x > \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 5$

Step 8.3.2

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

$- 2 {(5)}^{2} + 3 (5) + 5 > 0$

Step 8.3.3

The left side $- 30$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < - 1$ False

$- 1 < x < \frac{5}{2}$ True

$x > \frac{5}{2}$ False

$x < - 1$ False

$- 1 < x < \frac{5}{2}$ True

$x > \frac{5}{2}$ False

Step 9

The solution consists of all of the true intervals.

$- 1 < x < \frac{5}{2}$

Step 10

Convert the inequality to interval notation.

$(- 1, \frac{5}{2})$

Step 11