Calculus Examples

$12 x^{2} - 7 x - 12 < 0$

Step 1

Convert the inequality to an equation.

$12 x^{2} - 7 x - 12 = 0$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 12 \cdot - 12 = - 144$ and whose sum is $b = - 7$ .

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Step 2.1.1

Factor $- 7$ out of $- 7 x$ .

$12 x^{2} - 7 x - 12 = 0$

Step 2.1.2

Rewrite $- 7$ as $9$ plus $- 16$

$12 x^{2} + (9 - 16) x - 12 = 0$

Step 2.1.3

Apply the distributive property.

$12 x^{2} + 9 x - 16 x - 12 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(12 x^{2} + 9 x) - 16 x - 12 = 0$

Step 2.2.2

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

$3 x (4 x + 3) - 4 (4 x + 3) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$(4 x + 3) (3 x - 4) = 0$

Step 3

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

$4 x + 3 = 0$

$3 x - 4 = 0$

Step 4

Set $4 x + 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Step 4.2

Solve $4 x + 3 = 0$ for $x$ .

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Step 4.2.2

Divide each term in $4 x = - 3$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 5

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 5.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 5.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 5.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 6

The final solution is all the values that make $(4 x + 3) (3 x - 4) = 0$ true.

$x = - \frac{3}{4}, \frac{4}{3}$

Step 7

Use each root to create test intervals.

$x < - \frac{3}{4}$

$- \frac{3}{4} < x < \frac{4}{3}$

$x > \frac{4}{3}$

Step 8

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 8.1

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

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Step 8.1.1

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

$x = - 3$

Step 8.1.2

Replace $x$ with $- 3$ in the original inequality.

$12 {(- 3)}^{2} - 7 \cdot - 3 - 12 < 0$

Step 8.1.3

The left side $117$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.2

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

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Step 8.2.1

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

$x = 0$

Step 8.2.2

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

$12 {(0)}^{2} - 7 \cdot 0 - 12 < 0$

Step 8.2.3

The left side $- 12$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

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

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Step 8.3.1

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

$x = 4$

Step 8.3.2

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

$12 {(4)}^{2} - 7 \cdot 4 - 12 < 0$

Step 8.3.3

The left side $152$ is not less 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.

$x < - \frac{3}{4}$ False

$- \frac{3}{4} < x < \frac{4}{3}$ True

$x > \frac{4}{3}$ False

$x < - \frac{3}{4}$ False

$- \frac{3}{4} < x < \frac{4}{3}$ True

$x > \frac{4}{3}$ False

Step 9

The solution consists of all of the true intervals.

$- \frac{3}{4} < x < \frac{4}{3}$

Step 10

Convert the inequality to interval notation.

$(- \frac{3}{4}, \frac{4}{3})$

Step 11