Precalculus Examples

$\frac{- 5}{x - 7} \leq - 5$

Step 1

Add $5$ to both sides of the inequality.

$\frac{- 5}{x - 7} + 5 \leq 0$

Step 2

Simplify $\frac{- 5}{x - 7} + 5$ .

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Step 2.1

Move the negative in front of the fraction.

$- \frac{5}{x - 7} + 5 \leq 0$

Step 2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$- \frac{5}{x - 7} + \frac{5 (x - 7)}{x - 7} \leq 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{- 5 + 5 (x - 7)}{x - 7} \leq 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $5$ out of $- 5 + 5 (x - 7)$ .

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Step 2.4.1.1

Factor $5$ out of $- 5$ .

$\frac{5 \cdot - 1 + 5 (x - 7)}{x - 7} \leq 0$

Step 2.4.1.2

Factor $5$ out of $5 \cdot - 1 + 5 (x - 7)$ .

$\frac{5 (- 1 + x - 7)}{x - 7} \leq 0$

Step 2.4.2

Subtract $7$ from $- 1$ .

$\frac{5 (x - 8)}{x - 7} \leq 0$

Step 3

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

$x - 8 = 0$

$x - 7 = 0$

Step 4

Add $8$ to both sides of the equation.

$x = 8$

Step 5

Add $7$ to both sides of the equation.

$x = 7$

Step 6

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

$x = 8$

$x = 7$

Step 7

Consolidate the solutions.

$x = 8, 7$

Step 8

Find the domain of $\frac{5 (x - 8)}{x - 7}$ .

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Step 8.1

Set the denominator in $\frac{5 (x - 8)}{x - 7}$ equal to $0$ to find where the expression is undefined.

$x - 7 = 0$

Step 8.2

Add $7$ to both sides of the equation.

$x = 7$

Step 8.3

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

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

Step 9

Use each root to create test intervals.

$x < 7$

$7 < x < 8$

$x > 8$

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 $x < 7$ to see if it makes the inequality true.

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Step 10.1.1

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

$x = 0$

Step 10.1.2

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

$\frac{- 5}{(0) - 7} \leq - 5$

Step 10.1.3

The left side $0.\overline{714285}$ is greater than the right side $- 5$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $7 < x < 8$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $7 < x < 8$ and see if this value makes the original inequality true.

$x = 7.5$

Step 10.2.2

Replace $x$ with $7.5$ in the original inequality.

$\frac{- 5}{(7.5) - 7} \leq - 5$

Step 10.2.3

The left side $- 10$ is less than the right side $- 5$ , which means that the given statement is always true.

True

Step 10.3

Test a value on the interval $x > 8$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Step 10.3.2

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

$\frac{- 5}{(10) - 7} \leq - 5$

Step 10.3.3

The left side $- 1.\overline{6}$ is greater than the right side $- 5$ , which means that the given statement is false.

False

Step 10.4

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

$x < 7$ False

$7 < x < 8$ True

$x > 8$ False

$x < 7$ False

$7 < x < 8$ True

$x > 8$ False

Step 11

The solution consists of all of the true intervals.

$7 < x \leq 8$

Step 12

Convert the inequality to interval notation.

$(7, 8]$

Step 13