Solve -2≤4(n^3) 2 - Socratic AI (ScanMath)

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for n

n \geq - \frac{3 2}{2}

Alternative Form

n \in [- \frac{3 2}{2}, + \infty)

Evaluate

- 2 \leq 4 n^{3} \times 2

Multiply the terms

- 2 \leq 8 n^{3}

Move the expression to the left side

- 2 - 8 n^{3} \leq 0

Rewrite the expression

- 2 - 8 n^{3} = 0

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

- 8 n^{3} = 0 + 2

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

- 8 n^{3} = 2

Change the signs on both sides of the equation

8 n^{3} = - 2

Divide both sides

\frac{8 n ^{3}}{8} = \frac{- 2}{8}

Divide the numbers

n^{3} = \frac{- 2}{8}

Divide the numbers

More Steps

n^{3} = - \frac{1}{4}

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

3 n^{3} = 3 - \frac{1}{4}

Calculate

n = 3 - \frac{1}{4}

Simplify the root

More Steps

n = - \frac{3 2}{2}

Determine the test intervals using the critical values

n < - \frac{3 2}{2} n > - \frac{3 2}{2}

Choose a value form each interval

n_{1} = - 2 n_{2} = 0

To determine if n < - \frac{3 2}{2} is the solution to the inequality,test if the chosen value n = - 2 satisfies the initial inequality

More Steps

n < - \frac{3 2}{2} is not a solution n_{2} = 0

To determine if n > - \frac{3 2}{2} is the solution to the inequality,test if the chosen value n = 0 satisfies the initial inequality

More Steps

n < - \frac{3 2}{2} is not a solution n > - \frac{3 2}{2} is the solution

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

n \geq - \frac{3 2}{2} is the solution

Solution

n \geq - \frac{3 2}{2}

Alternative Form

n \in [- \frac{3 2}{2}, + \infty)

Show Solution