Precalculus Examples

$\frac{x - 4}{x - 7} > 0$

Step 1

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

$x - 4 = 0$

$x - 7 = 0$

Step 2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

Add $7$ to both sides of the equation.

$x = 7$

Step 4

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

$x = 4$

$x = 7$

Step 5

Consolidate the solutions.

$x = 4, 7$

Step 6

Find the domain of $\frac{x - 4}{x - 7}$ .

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Step 6.1

Set the denominator in $\frac{x - 4}{x - 7}$ equal to $0$ to find where the expression is undefined.

$x - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 6.3

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

$(- \infty, 7) \cup (7, \infty)$

Step 7

Use each root to create test intervals.

$x < 4$

$4 < x < 7$

$x > 7$

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 $x < 4$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 8.1.2

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

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

Step 8.1.3

The left side $0.\overline{571428}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval $4 < x < 7$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $4 < x < 7$ and see if this value makes the original inequality true.

$x = 6$

Step 8.2.2

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

$\frac{(6) - 4}{(6) - 7} > 0$

Step 8.2.3

The left side $- 2$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 8.3.2

Replace $x$ with $10$ in the original inequality.

$\frac{(10) - 4}{(10) - 7} > 0$

Step 8.3.3

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

True

Step 8.4

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

$x < 4$ True

$4 < x < 7$ False

$x > 7$ True

$x < 4$ True

$4 < x < 7$ False

$x > 7$ True

Step 9

The solution consists of all of the true intervals.

$x < 4$ or $x > 7$

Step 10

Convert the inequality to interval notation.

$(- \infty, 4) \cup (7, \infty)$

Step 11