Finite Math Examples

$\frac{x^{2} + | 3 x |}{x + 3} > 0$

Step 1

Remove non-negative terms from the absolute value.

$\frac{x^{2} + 3 | x |}{x + 3} > 0$

Step 2

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} + 3 | x | = 0$

$x + 3 = 0$

Step 3

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

$3 | x | = - x^{2}$

Step 4

Divide each term in $3 | x | = - x^{2}$ by $3$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $3 | x | = - x^{2}$ by $3$ .

$\frac{3 | x |}{3} = \frac{- x^{2}}{3}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{3 | x |}{3} = \frac{- x^{2}}{3}$

Step 4.2.1.2

Divide $| x |$ by $1$ .

$| x | = \frac{- x^{2}}{3}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Move the negative in front of the fraction.

$| x | = - \frac{x^{2}}{3}$

Step 5

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm (- \frac{x^{2}}{3})$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = - \frac{x^{2}}{3}$

Step 6.2

Multiply both sides by $3$ .

$x \cdot 3 = - \frac{x^{2}}{3} \cdot 3$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Simplify the left side.

Tap for more steps...

Step 6.3.1.1

Move $3$ to the left of $x$ .

$3 x = - \frac{x^{2}}{3} \cdot 3$

Step 6.3.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.2.1.1

Move the leading negative in $- \frac{x^{2}}{3}$ into the numerator.

$3 x = \frac{- x^{2}}{3} \cdot 3$

Step 6.3.2.1.2

Cancel the common factor.

$3 x = \frac{- x^{2}}{3} \cdot 3$

Step 6.3.2.1.3

Rewrite the expression.

$3 x = - x^{2}$

Step 6.4

Solve for $x$ .

Tap for more steps...

Step 6.4.1

Add $x^{2}$ to both sides of the equation.

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

Step 6.4.2

Factor the left side of the equation.

Tap for more steps...

Step 6.4.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$3 u + u^{2} = 0$

Step 6.4.2.2

Factor $u$ out of $3 u + u^{2}$ .

Tap for more steps...

Step 6.4.2.2.1

Factor $u$ out of $3 u$ .

$u \cdot 3 + u^{2} = 0$

Step 6.4.2.2.2

Factor $u$ out of $u^{2}$ .

$u \cdot 3 + u \cdot u = 0$

Step 6.4.2.2.3

Factor $u$ out of $u \cdot 3 + u \cdot u$ .

$u (3 + u) = 0$

Step 6.4.2.3

Replace all occurrences of $u$ with $x$ .

$x (3 + x) = 0$

Step 6.4.3

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

$x = 0$

$3 + x = 0$

Step 6.4.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.4.5

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

Tap for more steps...

Step 6.4.5.1

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

$3 + x = 0$

Step 6.4.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6.4.6

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

$x = 0, - 3$

Step 6.5

Next, use the negative value of the $\pm$ to find the second solution.

$x = \frac{x^{2}}{3}$

Step 6.6

Multiply both sides by $3$ .

$x \cdot 3 = \frac{x^{2}}{3} \cdot 3$

Step 6.7

Simplify.

Tap for more steps...

Step 6.7.1

Simplify the left side.

Tap for more steps...

Step 6.7.1.1

Move $3$ to the left of $x$ .

$3 x = \frac{x^{2}}{3} \cdot 3$

Step 6.7.2

Simplify the right side.

Tap for more steps...

Step 6.7.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.7.2.1.1

Cancel the common factor.

$3 x = \frac{x^{2}}{3} \cdot 3$

Step 6.7.2.1.2

Rewrite the expression.

$3 x = x^{2}$

Step 6.8

Solve for $x$ .

Tap for more steps...

Step 6.8.1

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

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

Step 6.8.2

Factor the left side of the equation.

Tap for more steps...

Step 6.8.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$3 u - u^{2} = 0$

Step 6.8.2.2

Factor $u$ out of $3 u - u^{2}$ .

Tap for more steps...

Step 6.8.2.2.1

Factor $u$ out of $3 u$ .

$u \cdot 3 - u^{2} = 0$

Step 6.8.2.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 3 + u (- u) = 0$

Step 6.8.2.2.3

Factor $u$ out of $u \cdot 3 + u (- u)$ .

$u (3 - u) = 0$

Step 6.8.2.3

Replace all occurrences of $u$ with $x$ .

$x (3 - x) = 0$

Step 6.8.3

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

$x = 0$

$3 - x = 0$

Step 6.8.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.8.5

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

Tap for more steps...

Step 6.8.5.1

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

$3 - x = 0$

Step 6.8.5.2

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

Tap for more steps...

Step 6.8.5.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 6.8.5.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Tap for more steps...

Step 6.8.5.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 6.8.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.8.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 6.8.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 6.8.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.8.5.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 6.8.6

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

$x = 0, 3$

Step 6.9

The complete solution is the result of both the positive and negative portions of the solution.

$x = 0, - 3, 3$

Step 7

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 8

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0, - 3, 3$

$x = - 3$

Step 9

Consolidate the solutions.

$x = 0, - 3, 3, - 3$

Step 10

Find the domain of $\frac{x^{2} + | 3 x |}{x + 3}$ .

Tap for more steps...

Step 10.1

Set the denominator in $\frac{x^{2} + | 3 x |}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 10.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 10.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 3) \cup (- 3, \infty)$

Step 11

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 0$

$0 < x < 3$

$x > 3$

Step 12

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 12.1

Test a value on the interval $x < - 3$ to see if it makes the inequality true.

Tap for more steps...

Step 12.1.1

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

$x = - 6$

Step 12.1.2

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

$\frac{{(- 6)}^{2} + | 3 (- 6) |}{(- 6) + 3} > 0$

Step 12.1.3

The left side $- 18$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $- 3 < x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 12.2.1

Choose a value on the interval $- 3 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 12.2.2

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

$\frac{{(- 2)}^{2} + | 3 (- 2) |}{(- 2) + 3} > 0$

Step 12.2.3

The left side $10$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $0 < x < 3$ to see if it makes the inequality true.

Tap for more steps...

Step 12.3.1

Choose a value on the interval $0 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 12.3.2

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

$\frac{{(2)}^{2} + | 3 (2) |}{(2) + 3} > 0$

Step 12.3.3

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

True

Step 12.4

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Tap for more steps...

Step 12.4.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 12.4.2

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

$\frac{{(6)}^{2} + | 3 (6) |}{(6) + 3} > 0$

Step 12.4.3

The left side $6$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 12.5

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

$x < - 3$ False

$- 3 < x < 0$ True

$0 < x < 3$ True

$x > 3$ True

$x < - 3$ False

$- 3 < x < 0$ True

$0 < x < 3$ True

$x > 3$ True

Step 13

The solution consists of all of the true intervals.

$- 3 < x < 0$ or $0 < x < 3$ or $x > 3$

Step 14

Convert the inequality to interval notation.

$(- 3, 0) \cup (0, 3) \cup (3, \infty)$

Step 15