Trigonometry Examples

$5 x - 2 < 2 x^{2}$

Step 1

Subtract $2 x^{2}$ from both sides of the inequality.

$5 x - 2 - 2 x^{2} < 0$

Step 2

Convert the inequality to an equation.

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

Step 3

Factor the left side of the equation.

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Step 3.1

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

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Step 3.1.1

Reorder the expression.

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Step 3.1.1.1

Move $- 2$ .

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

Step 3.1.1.2

Reorder $5 x$ and $- 2 x^{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Factor $- 1$ out of $5 x$ .

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

Step 3.1.4

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

Factor.

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Step 3.2.1

Factor by grouping.

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Step 3.2.1.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

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Step 3.2.1.1.1

Factor $- 5$ out of $- 5 x$ .

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

Step 3.2.1.1.2

Rewrite $- 5$ as $- 1$ plus $- 4$

$- (2 x^{2} + (- 1 - 4) x + 2) = 0$

Step 3.2.1.1.3

Apply the distributive property.

$- (2 x^{2} - 1 x - 4 x + 2) = 0$

Step 3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 x^{2} - 1 x) - 4 x + 2) = 0$

Step 3.2.1.2.2

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

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

Step 3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 3.2.2

Remove unnecessary parentheses.

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

Step 4

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

$2 x - 1 = 0$

$x - 2 = 0$

Step 5

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 5.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 5.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 5.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6

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

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Step 6.1

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

$x - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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

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

Step 8

Use each root to create test intervals.

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

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

$x > 2$

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

$x = 0$

Step 9.1.2

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

$5 (0) - 2 < 2 {(0)}^{2}$

Step 9.1.3

The left side $- 2$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 9.2

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

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Step 9.2.1

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

$x = 1$

Step 9.2.2

Replace $x$ with $1$ in the original inequality.

$5 (1) - 2 < 2 {(1)}^{2}$

Step 9.2.3

The left side $3$ is not less than the right side $2$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 9.3.2

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

$5 (4) - 2 < 2 {(4)}^{2}$

Step 9.3.3

The left side $18$ is less than the right side $32$ , which means that the given statement is always true.

True

Step 9.4

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

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

$\frac{1}{2} < x < 2$ False

$x > 2$ True

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

$\frac{1}{2} < x < 2$ False

$x > 2$ True

Step 10

The solution consists of all of the true intervals.

$x < \frac{1}{2}$ or $x > 2$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{1}{2} or x > 2$

Interval Notation:

$(- \infty, \frac{1}{2}) \cup (2, \infty)$

Step 12