Solve the Inequality for x (x+6)/(x+1)-2<0

$\frac{x + 6}{x + 1} - 2 < 0$

Step 1

Simplify $\frac{x + 6}{x + 1} - 2$ .

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x + 6}{x + 1} - 2 \cdot \frac{x + 1}{x + 1} < 0$

Simplify terms.

Combine $- 2$ and $\frac{x + 1}{x + 1}$ .

$\frac{x + 6}{x + 1} + \frac{- 2 (x + 1)}{x + 1} < 0$

Combine the numerators over the common denominator.

$\frac{x + 6 - 2 (x + 1)}{x + 1} < 0$

Simplify the numerator.

Apply the distributive property.

$\frac{x + 6 - 2 x - 2 \cdot 1}{x + 1} < 0$

Multiply $- 2$ by $1$ .

$\frac{x + 6 - 2 x - 2}{x + 1} < 0$

Subtract $2 x$ from $x$ .

$\frac{- x + 6 - 2}{x + 1} < 0$

Subtract $2$ from $6$ .

$\frac{- x + 4}{x + 1} < 0$

Simplify with factoring out.

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 4}{x + 1} < 0$

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{- (x) - 1 (- 4)}{x + 1} < 0$

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{- (x - 4)}{x + 1} < 0$

Simplify the expression.

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{- 1 (x - 4)}{x + 1} < 0$

Move the negative in front of the fraction.

$- \frac{x - 4}{x + 1} < 0$

Step 2

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

$x - 4 = 0$

$x + 1 = 0$

Step 3

Add $4$ to both sides of the equation.

$x = 4$

Step 4

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

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

$x = 4$

$x = - 1$

Step 6

Consolidate the solutions.

$x = 4, - 1$

Step 7

Find the domain of $\frac{x + 6}{x + 1} - 2$ .

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 8

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 4$

$x > 4$

Step 9

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

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

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.

$\frac{(- 4) + 6}{(- 4) + 1} - 2 < 0$

The left side $- 2.\overline{6}$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{(0) + 6}{(0) + 1} - 2 < 0$

The left side $4$ is not less than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 6$

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

$\frac{(6) + 6}{(6) + 1} - 2 < 0$

The left side $- 0.\overline{285714}$ is less than the right side $0$ , which means that the given statement is always true.

True

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

$x < - 1$ True

$- 1 < x < 4$ False

$x > 4$ True

$x < - 1$ True

$- 1 < x < 4$ False

$x > 4$ True

Step 10

The solution consists of all of the true intervals.

$x < - 1$ or $x > 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x > 4$

Interval Notation:

$(- \infty, - 1) \cup (4, \infty)$

Step 12

Solve the Inequality for x (x+6)/(x+1)-2<0

Related Questions

Support LCMGCF.com by disabling your adblocker.