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

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

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

$x + 4 = 0$

$2 x - 1 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Add $1$ to both sides of the equation.

$2 x = 1$

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 $1$ .

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

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

$x = - 4$

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

Consolidate the solutions.

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

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

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

$2 x - 1 = 0$

Solve for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

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 $1$ .

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

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

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

Use each root to create test intervals.

$x < - 4$

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

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

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.

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

The left side $0.\overline{153846}$ is not less than the right side $0$ , which means that the given statement is false.

False

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.

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

The left side $- 4$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{1}{2}$ 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.

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

The left side $1.4$ is not less 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.

$x < - 4$ False

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

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

$x < - 4$ False

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

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

The solution consists of all of the true intervals.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

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

Related Questions

Support LCMGCF.com by disabling your adblocker.