Convert to Interval Notation -7/(x+5)<=-8/(x+6)

$- \frac{7}{x + 5} \leq - \frac{8}{x + 6}$

Step 1

Add $\frac{8}{x + 6}$ to both sides of the inequality.

$- \frac{7}{x + 5} + \frac{8}{x + 6} \leq 0$

Step 2

Simplify $- \frac{7}{x + 5} + \frac{8}{x + 6}$ .

To write $- \frac{7}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$- \frac{7}{x + 5} \cdot \frac{x + 6}{x + 6} + \frac{8}{x + 6} \leq 0$

To write $\frac{8}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$- \frac{7}{x + 5} \cdot \frac{x + 6}{x + 6} + \frac{8}{x + 6} \cdot \frac{x + 5}{x + 5} \leq 0$

Write each expression with a common denominator of $(x + 5) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{x + 5}$ by $\frac{x + 6}{x + 6}$ .

$- \frac{7 (x + 6)}{(x + 5) (x + 6)} + \frac{8}{x + 6} \cdot \frac{x + 5}{x + 5} \leq 0$

Multiply $\frac{8}{x + 6}$ by $\frac{x + 5}{x + 5}$ .

$- \frac{7 (x + 6)}{(x + 5) (x + 6)} + \frac{8 (x + 5)}{(x + 6) (x + 5)} \leq 0$

Reorder the factors of $(x + 6) (x + 5)$ .

$- \frac{7 (x + 6)}{(x + 5) (x + 6)} + \frac{8 (x + 5)}{(x + 5) (x + 6)} \leq 0$

Combine the numerators over the common denominator.

$\frac{- 7 (x + 6) + 8 (x + 5)}{(x + 5) (x + 6)} \leq 0$

Simplify the numerator.

Apply the distributive property.

$\frac{- 7 x - 7 \cdot 6 + 8 (x + 5)}{(x + 5) (x + 6)} \leq 0$

Multiply $- 7$ by $6$ .

$\frac{- 7 x - 42 + 8 (x + 5)}{(x + 5) (x + 6)} \leq 0$

Apply the distributive property.

$\frac{- 7 x - 42 + 8 x + 8 \cdot 5}{(x + 5) (x + 6)} \leq 0$

Multiply $8$ by $5$ .

$\frac{- 7 x - 42 + 8 x + 40}{(x + 5) (x + 6)} \leq 0$

Add $- 7 x$ and $8 x$ .

$\frac{x - 42 + 40}{(x + 5) (x + 6)} \leq 0$

Add $- 42$ and $40$ .

$\frac{x - 2}{(x + 5) (x + 6)} \leq 0$

Step 3

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

$x - 2 = 0$

$x + 5 = 0$

$x + 6 = 0$

Step 4

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 7

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

$x = 2$

$x = - 5$

$x = - 6$

Step 8

Consolidate the solutions.

$x = 2, - 5, - 6$

Step 9

Find the domain of $\frac{x - 2}{(x + 5) (x + 6)}$ .

Set the denominator in $\frac{x - 2}{(x + 5) (x + 6)}$ equal to $0$ to find where the expression is undefined.

$(x + 5) (x + 6) = 0$

Solve for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 5 = 0$

$x + 6 = 0$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x + 5) (x + 6) = 0$ true.

$x = - 5, - 6$

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

$(- \infty, - 6) \cup (- 6, - 5) \cup (- 5, \infty)$

Step 10

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - 5$

$- 5 < x < 2$

$x > 2$

Step 11

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.

$- \frac{7}{(- 8) + 5} \leq - \frac{8}{(- 8) + 6}$

The left side $2.\overline{3}$ is less than the right side $4$ , which means that the given statement is always true.

True

Test a value on the interval $- 6 < x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < - 5$ and see if this value makes the original inequality true.

$x = - 5.5$

Replace $x$ with $- 5.5$ in the original inequality.

$- \frac{7}{(- 5.5) + 5} \leq - \frac{8}{(- 5.5) + 6}$

The left side $14$ is greater than the right side $- 16$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$- \frac{7}{(0) + 5} \leq - \frac{8}{(0) + 6}$

The left side $- 1.4$ is less than the right side $- 1.\overline{3}$ , which means that the given statement is always true.

True

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

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

$- \frac{7}{(4) + 5} \leq - \frac{8}{(4) + 6}$

The left side $- 0.\overline{7}$ is greater than the right side $- 0.8$ , which means that the given statement is false.

False

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

$x < - 6$ True

$- 6 < x < - 5$ False

$- 5 < x < 2$ True

$x > 2$ False

$x < - 6$ True

$- 6 < x < - 5$ False

$- 5 < x < 2$ True

$x > 2$ False

Step 12

The solution consists of all of the true intervals.

$x < - 6$ or $- 5 < x \leq 2$

Step 13

Convert the inequality to interval notation.

$(- \infty, - 6) \cup (- 5, 2]$

Step 14

Convert to Interval Notation -7/(x+5)<=-8/(x+6)

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