Convert to Interval Notation (2x+5)/(8x-3)<=0

$\frac{2 x + 5}{8 x - 3} \leq 0$

Step 1

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

$2 x + 5 = 0$

$8 x - 3 = 0$

Step 2

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Step 3

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 4

Add $3$ to both sides of the equation.

$8 x = 3$

Step 5

Divide each term in $8 x = 3$ by $8$ and simplify.

Divide each term in $8 x = 3$ by $8$ .

$\frac{8 x}{8} = \frac{3}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{3}{8}$

Divide $x$ by $1$ .

$x = \frac{3}{8}$

Step 6

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

$x = - \frac{5}{2}$

$x = \frac{3}{8}$

Step 7

Consolidate the solutions.

$x = - \frac{5}{2}, \frac{3}{8}$

Step 8

Find the domain of $\frac{2 x + 5}{8 x - 3}$ .

Set the denominator in $\frac{2 x + 5}{8 x - 3}$ equal to $0$ to find where the expression is undefined.

$8 x - 3 = 0$

Solve for $x$ .

Add $3$ to both sides of the equation.

$8 x = 3$

Divide each term in $8 x = 3$ by $8$ and simplify.

Divide each term in $8 x = 3$ by $8$ .

$\frac{8 x}{8} = \frac{3}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{3}{8}$

Divide $x$ by $1$ .

$x = \frac{3}{8}$

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

$(- \infty, \frac{3}{8}) \cup (\frac{3}{8}, \infty)$

Step 9

Use each root to create test intervals.

$x < - \frac{5}{2}$

$- \frac{5}{2} < x < \frac{3}{8}$

$x > \frac{3}{8}$

Step 10

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 $x < - \frac{5}{2}$ to see if it makes the inequality true.

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

$x = - 5$

Replace $x$ with $- 5$ in the original inequality.

$\frac{2 (- 5) + 5}{8 (- 5) - 3} \leq 0$

The left side $0.11627906$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{5}{2} < x < \frac{3}{8}$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{5}{2} < x < \frac{3}{8}$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{2 (0) + 5}{8 (0) - 3} \leq 0$

The left side $- 1.\overline{6}$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{3}{8}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{3}{8}$ and see if this value makes the original inequality true.

$x = 3$

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

$\frac{2 (3) + 5}{8 (3) - 3} \leq 0$

The left side $0.\overline{523809}$ 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.

$x < - \frac{5}{2}$ False

$- \frac{5}{2} < x < \frac{3}{8}$ True

$x > \frac{3}{8}$ False

$x < - \frac{5}{2}$ False

$- \frac{5}{2} < x < \frac{3}{8}$ True

$x > \frac{3}{8}$ False

Step 11

The solution consists of all of the true intervals.

$- \frac{5}{2} \leq x < \frac{3}{8}$

Step 12

Convert the inequality to interval notation.

$[- \frac{5}{2}, \frac{3}{8})$

Step 13

Convert to Interval Notation (2x+5)/(8x-3)<=0

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