Precalculus Examples

$\frac{x - 5}{x - 4} \leq 0$

Step 1

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

$x - 5 = 0$

$x - 4 = 0$

Step 2

Add $5$ to both sides of the equation.

$x = 5$

Step 3

Add $4$ to both sides of the equation.

$x = 4$

Step 4

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

$x = 5$

$x = 4$

Step 5

Consolidate the solutions.

$x = 5, 4$

Step 6

Find the domain of $\frac{x - 5}{x - 4}$ .

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Step 6.1

Set the denominator in $\frac{x - 5}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6.3

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

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

Step 7

Use each root to create test intervals.

$x < 4$

$4 < x < 5$

$x > 5$

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.

$\frac{(0) - 5}{(0) - 4} \leq 0$

Step 8.1.3

The left side $1.25$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $4 < x < 5$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $4 < x < 5$ and see if this value makes the original inequality true.

$x = 4.5$

Step 8.2.2

Replace $x$ with $4.5$ in the original inequality.

$\frac{(4.5) - 5}{(4.5) - 4} \leq 0$

Step 8.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 8.3.2

Replace $x$ with $8$ in the original inequality.

$\frac{(8) - 5}{(8) - 4} \leq 0$

Step 8.3.3

The left side $0.75$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < 4$ False

$4 < x < 5$ True

$x > 5$ False

$x < 4$ False

$4 < x < 5$ True

$x > 5$ False

Step 9

The solution consists of all of the true intervals.

$4 < x \leq 5$

Step 10

Convert the inequality to interval notation.

$(4, 5]$

Step 11