Solve (-4-2x)/x<=0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, - 2] \cup (0, + \infty)

Evaluate

\frac{- 4 - 2 x}{x} \leq 0

Find the domain

\frac{- 4 - 2 x}{x} \leq 0, x \neq = 0

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{4 + 2 x}{x} \leq 0

Change the signs on both sides of the inequality and flip the inequality sign

\frac{4 + 2 x}{x} \geq 0

Set the numerator and denominator of \frac{4 + 2 x}{x} equal to 0 to find the values of x where sign changes may occur

4 + 2 x = 0 x = 0

Calculate

More Steps

x = - 2 x = 0

Determine the test intervals using the critical values

x < - 2 - 2 < x < 0 x > 0

Choose a value form each interval

x_{1} = - 3 x_{2} = - 1 x_{3} = 1

To determine if x < - 2 is the solution to the inequality,test if the chosen value x = - 3 satisfies the initial inequality

More Steps

x < - 2 is the solution x_{2} = - 1 x_{3} = 1

To determine if - 2 < x < 0 is the solution to the inequality,test if the chosen value x = - 1 satisfies the initial inequality

More Steps

x < - 2 is the solution - 2 < x < 0 is not a solution x_{3} = 1

To determine if x > 0 is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < - 2 is the solution - 2 < x < 0 is not a solution x > 0 is the solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

x \leq - 2 is the solution x > 0 is the solution

The final solution of the original inequality is x \in (- \infty, - 2] \cup (0, + \infty)

x \in (- \infty, - 2] \cup (0, + \infty)

Check if the solution is in the defined range

x \in (- \infty, - 2] \cup (0, + \infty), x \neq = 0

Solution

x \in (- \infty, - 2] \cup (0, + \infty)

Show Solution