Calculus Examples

$2 x^{2} + 1 < 9 x - 3$

Step 1

Subtract $9 x$ from both sides of the inequality.

$2 x^{2} + 1 - 9 x < - 3$

Step 2

Convert the inequality to an equation.

$2 x^{2} + 1 - 9 x = - 3$

Step 3

Add $3$ to both sides of the equation.

$2 x^{2} + 1 - 9 x + 3 = 0$

Step 4

Add $1$ and $3$ .

$2 x^{2} - 9 x + 4 = 0$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 2 \cdot 4 = 8$ and whose sum is $b = - 9$ .

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Step 5.1.1

Factor $- 9$ out of $- 9 x$ .

$2 x^{2} - 9 x + 4 = 0$

Step 5.1.2

Rewrite $- 9$ as $- 1$ plus $- 8$

$2 x^{2} + (- 1 - 8) x + 4 = 0$

Step 5.1.3

Apply the distributive property.

$2 x^{2} - 1 x - 8 x + 4 = 0$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$(2 x^{2} - 1 x) - 8 x + 4 = 0$

Step 5.2.2

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

$x (2 x - 1) - 4 (2 x - 1) = 0$

Step 5.3

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

$(2 x - 1) (x - 4) = 0$

Step 6

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

$2 x - 1 = 0$

$x - 4 = 0$

Step 7

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

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Step 7.1

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

$2 x - 1 = 0$

Step 7.2

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

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Step 7.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 8

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 9

The final solution is all the values that make $(2 x - 1) (x - 4) = 0$ true.

$x = \frac{1}{2}, 4$

Step 10

Use each root to create test intervals.

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

$\frac{1}{2} < x < 4$

$x > 4$

Step 11

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

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Step 11.1

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

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Step 11.1.1

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

$x = 0$

Step 11.1.2

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

$2 {(0)}^{2} + 1 < 9 (0) - 3$

Step 11.1.3

The left side $1$ is not less than the right side $- 3$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $\frac{1}{2} < x < 4$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $\frac{1}{2} < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 11.2.2

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

$2 {(2)}^{2} + 1 < 9 (2) - 3$

Step 11.2.3

The left side $9$ is less than the right side $15$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 11.3.2

Replace $x$ with $6$ in the original inequality.

$2 {(6)}^{2} + 1 < 9 (6) - 3$

Step 11.3.3

The left side $73$ is not less than the right side $51$ , which means that the given statement is false.

False

Step 11.4

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

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

$\frac{1}{2} < x < 4$ True

$x > 4$ False

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

$\frac{1}{2} < x < 4$ True

$x > 4$ False

Step 12

The solution consists of all of the true intervals.

$\frac{1}{2} < x < 4$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$\frac{1}{2} < x < 4$

Interval Notation:

$(\frac{1}{2}, 4)$

Step 14