Finite Math Examples

$3 x (x - 1) + 2 x > 12 - x$

Step 1

Simplify $3 x (x - 1) + 2 x$ .

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 1 + 2 x > 12 - x$

Step 1.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.2.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 1 + 2 x > 12 - x$

Step 1.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 1 + 2 x > 12 - x$

Step 1.1.3

Multiply $- 1$ by $3$ .

$3 x^{2} - 3 x + 2 x > 12 - x$

Step 1.2

Add $- 3 x$ and $2 x$ .

$3 x^{2} - x > 12 - x$

Step 2

Move all terms containing $x$ to the left side of the inequality.

Tap for more steps...

Step 2.1

Add $x$ to both sides of the inequality.

$3 x^{2} - x + x > 12$

Step 2.2

Combine the opposite terms in $3 x^{2} - x + x$ .

Tap for more steps...

Step 2.2.1

Add $- x$ and $x$ .

$3 x^{2} + 0 > 12$

Step 2.2.2

Add $3 x^{2}$ and $0$ .

$3 x^{2} > 12$

Step 3

Divide each term in $3 x^{2} > 12$ by $3$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $3 x^{2} > 12$ by $3$ .

$\frac{3 x^{2}}{3} > \frac{12}{3}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} > \frac{12}{3}$

Step 3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{12}{3}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Divide $12$ by $3$ .

$x^{2} > 4$

Step 4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} > \sqrt{4}$

Step 5

Simplify the equation.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Pull terms out from under the radical.

$| x | > \sqrt{4}$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $\sqrt{4}$ .

Tap for more steps...

Step 5.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | > \sqrt{2^{2}}$

Step 5.2.1.2

Pull terms out from under the radical.

$| x | > | 2 |$

Step 5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | > 2$

Step 6

Write $| x | > 2$ as a piecewise.

Tap for more steps...

Step 6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 2$

Step 6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 2$

Step 6.5

Write as a piecewise.

${\begin{cases} x > 2 & x \geq 0 \\ - x > 2 & x < 0 \end{cases}$

Step 7

Find the intersection of $x > 2$ and $x \geq 0$ .

$x > 2$

Step 8

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

Tap for more steps...

Step 8.1

Divide each term in $- x > 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Dividing two negative values results in a positive value.

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

Step 8.2.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Divide $2$ by $- 1$ .

$x < - 2$

Step 9

Find the union of the solutions.

$x < - 2$ or $x > 2$

Step 10

Convert the inequality to interval notation.

$(- \infty, - 2) \cup (2, \infty)$

Step 11