Solve -2(x^3) ≤ 2 - x^3

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

x \geq - 32

Alternative Form

x \in [- 32, + \infty)

Evaluate

- 2 x^{3} \leq 2 - x^{3}

Move the expression to the left side

- 2 x^{3} - (2 - x^{3}) \leq 0

Subtract the terms

More Steps

- x^{3} - 2 \leq 0

Rewrite the expression

- x^{3} - 2 = 0

Move the constant to the right-hand side and change its sign

- x^{3} = 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

- x^{3} = 2

Change the signs on both sides of the equation

x^{3} = - 2

Take the 3 -th root on both sides of the equation

3 x^{3} = 3 - 2

Calculate

x = 3 - 2

An odd root of a negative radicand is always a negative

x = - 32

Determine the test intervals using the critical values

x < - 32 x > - 32

Choose a value form each interval

x_{1} = - 2 x_{2} = 0

To determine if x < - 32 is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < - 32 is not a solution x_{2} = 0

To determine if x > - 32 is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

More Steps

x < - 32 is not a solution x > - 32 is the solution

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

x \geq - 32 is the solution

Solution

x \geq - 32

Alternative Form

x \in [- 32, + \infty)

Show Solution