Graph (x+6)/(6-x)<0

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

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

$x + 6 = 0$

$6 - x = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Subtract $6$ from both sides of the equation.

$- x = - 6$

Divide each term in $- x = - 6$ by $- 1$ and simplify.

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x = 6$

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

$x = - 6$

$x = 6$

Consolidate the solutions.

$x = - 6, 6$

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

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

$6 - x = 0$

Solve for $x$ .

Subtract $6$ from both sides of the equation.

$- x = - 6$

Divide each term in $- x = - 6$ by $- 1$ and simplify.

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x = 6$

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

$(- \infty, 6) \cup (6, \infty)$

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 6$

$x > 6$

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 $x < - 6$ to see if it makes the inequality true.

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.

$\frac{(- 8) + 6}{6 - (- 8)} < 0$

The left side $- 0.\overline{142857}$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

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

The left side $1$ is not less than the right side $0$ , which means that the given statement is false.

False

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

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.

$\frac{(8) + 6}{6 - (8)} < 0$

The left side $- 7$ 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.

$x < - 6$ True

$- 6 < x < 6$ False

$x > 6$ True

$x < - 6$ True

$- 6 < x < 6$ False

$x > 6$ True

The solution consists of all of the true intervals.

$x < - 6$ or $x > 6$

Graph (x+6)/(6-x)<0

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