Graph (x-9)/(7x+2)<=0

$\frac{x - 9}{7 x + 2} \leq 0$

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

$x - 9 = 0$

$7 x + 2 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Subtract $2$ from both sides of the equation.

$7 x = - 2$

Divide each term in $7 x = - 2$ by $7$ and simplify.

Divide each term in $7 x = - 2$ by $7$ .

$\frac{7 x}{7} = \frac{- 2}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 2}{7}$

Divide $x$ by $1$ .

$x = \frac{- 2}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{7}$

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

$x = 9$

$x = - \frac{2}{7}$

Consolidate the solutions.

$x = 9, - \frac{2}{7}$

Find the domain of $\frac{x - 9}{7 x + 2}$ .

Set the denominator in $\frac{x - 9}{7 x + 2}$ equal to $0$ to find where the expression is undefined.

$7 x + 2 = 0$

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$7 x = - 2$

Divide each term in $7 x = - 2$ by $7$ and simplify.

Divide each term in $7 x = - 2$ by $7$ .

$\frac{7 x}{7} = \frac{- 2}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 2}{7}$

Divide $x$ by $1$ .

$x = \frac{- 2}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{7}$

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

$(- \infty, - \frac{2}{7}) \cup (- \frac{2}{7}, \infty)$

Use each root to create test intervals.

$x < - \frac{2}{7}$

$- \frac{2}{7} < x < 9$

$x > 9$

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 $x < - \frac{2}{7}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{2}{7}$ and see if this value makes the original inequality true.

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

$\frac{(- 3) - 9}{7 (- 3) + 2} \leq 0$

The left side $0.63157894$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{2}{7} < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{2}{7} < x < 9$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{(0) - 9}{7 (0) + 2} \leq 0$

The left side $- 4.5$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$\frac{(12) - 9}{7 (12) + 2} \leq 0$

The left side $0.03488372$ is greater than the right side $0$ , which means that the given statement is false.

False

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

$x < - \frac{2}{7}$ False

$- \frac{2}{7} < x < 9$ True

$x > 9$ False

$x < - \frac{2}{7}$ False

$- \frac{2}{7} < x < 9$ True

$x > 9$ False

The solution consists of all of the true intervals.

$- \frac{2}{7} < x \leq 9$

Graph (x-9)/(7x+2)<=0

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