Finite Math Examples

$- \frac{4 (2 x - 2)}{3 x} > 6$

Step 1

Subtract $6$ from both sides of the inequality.

$- \frac{4 (2 x - 2)}{3 x} - 6 > 0$

Step 2

Simplify $- \frac{4 (2 x - 2)}{3 x} - 6$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Factor $2$ out of $2 x - 2$ .

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Step 2.1.1.1

Factor $2$ out of $2 x$ .

$- \frac{4 (2 (x) - 2)}{3 x} - 6 > 0$

Step 2.1.1.2

Factor $2$ out of $- 2$ .

$- \frac{4 (2 (x) + 2 (- 1))}{3 x} - 6 > 0$

Step 2.1.1.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$- \frac{4 (2 (x - 1))}{3 x} - 6 > 0$

$- \frac{4 \cdot 2 (x - 1)}{3 x} - 6 > 0$

Step 2.1.2

Multiply $4$ by $2$ .

$- \frac{8 (x - 1)}{3 x} - 6 > 0$

Step 2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3 x}{3 x}$ .

$- \frac{8 (x - 1)}{3 x} - 6 \cdot \frac{3 x}{3 x} > 0$

Step 2.3

Combine $- 6$ and $\frac{3 x}{3 x}$ .

$- \frac{8 (x - 1)}{3 x} + \frac{- 6 (3 x)}{3 x} > 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- 8 (x - 1) - 6 (3 x)}{3 x} > 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $2$ out of $- 8 (x - 1) - 6 \cdot 3 x$ .

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Step 2.5.1.1

Factor $2$ out of $- 8 (x - 1)$ .

$\frac{2 (- 4 (x - 1)) - 6 \cdot 3 x}{3 x} > 0$

Step 2.5.1.2

Factor $2$ out of $- 6 \cdot 3 x$ .

$\frac{2 (- 4 (x - 1)) + 2 (- 3 \cdot 3 x)}{3 x} > 0$

Step 2.5.1.3

Factor $2$ out of $2 (- 4 (x - 1)) + 2 (- 3 \cdot 3 x)$ .

$\frac{2 (- 4 (x - 1) - 3 \cdot 3 x)}{3 x} > 0$

Step 2.5.2

Apply the distributive property.

$\frac{2 (- 4 x - 4 \cdot - 1 - 3 \cdot 3 x)}{3 x} > 0$

Step 2.5.3

Multiply $- 4$ by $- 1$ .

$\frac{2 (- 4 x + 4 - 3 \cdot 3 x)}{3 x} > 0$

Step 2.5.4

Multiply $- 3$ by $3$ .

$\frac{2 (- 4 x + 4 - 9 x)}{3 x} > 0$

Step 2.5.5

Subtract $9 x$ from $- 4 x$ .

$\frac{2 (- 13 x + 4)}{3 x} > 0$

Step 2.6

Factor $- 1$ out of $- 13 x$ .

$\frac{2 (- (13 x) + 4)}{3 x} > 0$

Step 2.7

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{2 (- (13 x) - 1 (- 4))}{3 x} > 0$

Step 2.8

Factor $- 1$ out of $- (13 x) - 1 (- 4)$ .

$\frac{2 (- (13 x - 4))}{3 x} > 0$

Step 2.9

Rewrite $- (13 x - 4)$ as $- 1 (13 x - 4)$ .

$\frac{2 (- 1 (13 x - 4))}{3 x} > 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{2 (13 x - 4)}{3 x} > 0$

Step 3

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

$3 x = 0$

$13 x - 4 = 0$

Step 4

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $3$ .

$x = 0$

Step 5

Add $4$ to both sides of the equation.

$13 x = 4$

Step 6

Divide each term in $13 x = 4$ by $13$ and simplify.

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Step 6.1

Divide each term in $13 x = 4$ by $13$ .

$\frac{13 x}{13} = \frac{4}{13}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $13$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{4}{13}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{13}$

Step 7

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

$x = 0$

$x = \frac{4}{13}$

Step 8

Consolidate the solutions.

$x = 0, \frac{4}{13}$

Step 9

Find the domain of $- \frac{2 (13 x - 4)}{3 x}$ .

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Step 9.1

Set the denominator in $\frac{2 (13 x - 4)}{3 x}$ equal to $0$ to find where the expression is undefined.

$3 x = 0$

Step 9.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 9.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $3$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 9.3

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

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

Step 10

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{4}{13}$

$x > \frac{4}{13}$

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 $x < 0$ to see if it makes the inequality true.

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Step 11.1.1

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

$x = - 2$

Step 11.1.2

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

$- \frac{4 (2 (- 2) - 2)}{3 (- 2)} > 6$

Step 11.1.3

The left side $- 4$ is not greater than the right side $6$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $0 < x < \frac{4}{13}$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $0 < x < \frac{4}{13}$ and see if this value makes the original inequality true.

$x = 0.15$

Step 11.2.2

Replace $x$ with $0.15$ in the original inequality.

$- \frac{4 (2 (0.15) - 2)}{3 (0.15)} > 6$

Step 11.2.3

The left side $15.\overline{1}$ is greater than the right side $6$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $x > \frac{4}{13}$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > \frac{4}{13}$ and see if this value makes the original inequality true.

$x = 3$

Step 11.3.2

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

$- \frac{4 (2 (3) - 2)}{3 (3)} > 6$

Step 11.3.3

The left side $- 1.\overline{7}$ is not greater than the right side $6$ , which means that the given statement is false.

False

Step 11.4

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

$x < 0$ False

$0 < x < \frac{4}{13}$ True

$x > \frac{4}{13}$ False

$x < 0$ False

$0 < x < \frac{4}{13}$ True

$x > \frac{4}{13}$ False

Step 12

The solution consists of all of the true intervals.

$0 < x < \frac{4}{13}$

Step 13

Convert the inequality to interval notation.

$(0, \frac{4}{13})$

Step 14