Calculus Examples

$\frac{2 x - 5}{x + 1} \leq 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{2 x - 5}{x + 1} - 1 \leq 0$

Step 2

Simplify $\frac{2 x - 5}{x + 1} - 1$ .

Tap for more steps...

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{2 x - 5}{x + 1} - 1 \cdot \frac{x + 1}{x + 1} \leq 0$

Combine $- 1$ and $\frac{x + 1}{x + 1}$ .

$\frac{2 x - 5}{x + 1} + \frac{- (x + 1)}{x + 1} \leq 0$

Combine the numerators over the common denominator.

$\frac{2 x - 5 - (x + 1)}{x + 1} \leq 0$

Simplify the numerator.

Tap for more steps...

Apply the distributive property.

$\frac{2 x - 5 - x - 1 \cdot 1}{x + 1} \leq 0$

Multiply $- 1$ by $1$ .

$\frac{2 x - 5 - x - 1}{x + 1} \leq 0$

Subtract $x$ from $2 x$ .

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

Subtract $1$ from $- 5$ .

$\frac{x - 6}{x + 1} \leq 0$

Step 3

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

$x - 6 = 0$

$x + 1 = 0$

Step 4

Add $6$ to both sides of the equation.

$x = 6$

Step 5

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

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

$x = 6$

$x = - 1$

Step 7

Consolidate the solutions.

$x = 6, - 1$

Step 8

Find the domain of $\frac{x - 6}{x + 1}$ .

Tap for more steps...

Set the denominator in $\frac{x - 6}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$(- \infty, - 1) \cup (- 1, \infty)$

Step 9

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 6$

$x > 6$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Tap for more steps...

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

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

$\frac{2 (- 4) - 5}{(- 4) + 1} \leq 1$

The left side $4.\overline{3}$ is greater than the right side $1$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 6$ to see if it makes the inequality true.

Tap for more steps...

Choose a value on the interval $- 1 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{2 (0) - 5}{(0) + 1} \leq 1$

The left side $- 5$ is less than the right side $1$ , which means that the given statement is always true.

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Tap for more steps...

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

$\frac{2 (8) - 5}{(8) + 1} \leq 1$

The left side $1.\overline{2}$ is greater than the right side $1$ , which means that the given statement is false.

False

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

$x < - 1$ False

$- 1 < x < 6$ True

$x > 6$ False

$x < - 1$ False

$- 1 < x < 6$ True

$x > 6$ False

Step 11

The solution consists of all of the true intervals.

$- 1 < x \leq 6$

Step 12

Convert the inequality to interval notation.

$(- 1, 6]$

Step 13