Solve the Inequality for x (2x-1)/(x+4)>=0

$\frac{2 x - 1}{x + 4} \geq 0$

Step 1

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

$2 x - 1 = 0$

$x + 4 = 0$

Step 2

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 4

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

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

$x = \frac{1}{2}$

$x = - 4$

Step 6

Consolidate the solutions.

$x = \frac{1}{2}, - 4$

Step 7

Find the domain of $\frac{2 x - 1}{x + 4}$ .

Set the denominator in $\frac{2 x - 1}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

Step 8

Use each root to create test intervals.

$x < - 4$

$- 4 < x < \frac{1}{2}$

$x > \frac{1}{2}$

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 $x < - 4$ 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.

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

The left side $6.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < \frac{1}{2}$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

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

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

The left side $- 0.25$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{1}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 3$

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

$\frac{2 (3) - 1}{(3) + 4} \geq 0$

The left side $0.\overline{714285}$ is greater 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.

$x < - 4$ True

$- 4 < x < \frac{1}{2}$ False

$x > \frac{1}{2}$ True

$x < - 4$ True

$- 4 < x < \frac{1}{2}$ False

$x > \frac{1}{2}$ True

Step 10

The solution consists of all of the true intervals.

$x < - 4$ or $x \geq \frac{1}{2}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - 4 or x \geq \frac{1}{2}$

Interval Notation:

$(- \infty, - 4) \cup [\frac{1}{2}, \infty)$

Step 12

Solve the Inequality for x (2x-1)/(x+4)>=0

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