Finite Math Examples

$\frac{2 x + 3}{3 x - 7} > 0$

Step 1

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

$2 x + 3 = 0$

$3 x - 7 = 0$

Step 2

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 3

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x = - 3$ by $2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

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

Step 4

Add $7$ to both sides of the equation.

$3 x = 7$

Step 5

Divide each term in $3 x = 7$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Step 6

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

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

$x = \frac{7}{3}$

Step 7

Consolidate the solutions.

$x = - \frac{3}{2}, \frac{7}{3}$

Step 8

Find the domain of $\frac{2 x + 3}{3 x - 7}$ .

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Step 8.1

Set the denominator in $\frac{2 x + 3}{3 x - 7}$ equal to $0$ to find where the expression is undefined.

$3 x - 7 = 0$

Step 8.2

Solve for $x$ .

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Step 8.2.1

Add $7$ to both sides of the equation.

$3 x = 7$

Step 8.2.2

Divide each term in $3 x = 7$ by $3$ and simplify.

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Step 8.2.2.1

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Step 8.3

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

$(- \infty, \frac{7}{3}) \cup (\frac{7}{3}, \infty)$

Step 9

Use each root to create test intervals.

$x < - \frac{3}{2}$

$- \frac{3}{2} < x < \frac{7}{3}$

$x > \frac{7}{3}$

Step 10

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 10.1

Test a value on the interval $x < - \frac{3}{2}$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $x < - \frac{3}{2}$ and see if this value makes the original inequality true.

$x = - 4$

Step 10.1.2

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

$\frac{2 (- 4) + 3}{3 (- 4) - 7} > 0$

Step 10.1.3

The left side $0.26315789$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- \frac{3}{2} < x < \frac{7}{3}$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- \frac{3}{2} < x < \frac{7}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.2.2

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

$\frac{2 (0) + 3}{3 (0) - 7} > 0$

Step 10.2.3

The left side $- 0.\overline{428571}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $x > \frac{7}{3}$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > \frac{7}{3}$ and see if this value makes the original inequality true.

$x = 5$

Step 10.3.2

Replace $x$ with $5$ in the original inequality.

$\frac{2 (5) + 3}{3 (5) - 7} > 0$

Step 10.3.3

The left side $1.625$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.4

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

$x < - \frac{3}{2}$ True

$- \frac{3}{2} < x < \frac{7}{3}$ False

$x > \frac{7}{3}$ True

$x < - \frac{3}{2}$ True

$- \frac{3}{2} < x < \frac{7}{3}$ False

$x > \frac{7}{3}$ True

Step 11

The solution consists of all of the true intervals.

$x < - \frac{3}{2}$ or $x > \frac{7}{3}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{3}{2} or x > \frac{7}{3}$

Interval Notation:

$(- \infty, - \frac{3}{2}) \cup (\frac{7}{3}, \infty)$

Step 13