Solve |x^2-4x^3|<3

Question

Solve the inequality

- 0.832406 < x < 1

Alternative Form

x \in (- 0.832406, 1)

Evaluate

x^{2} - 4 x^{3} < 3

Separate the inequality into 2 possible cases

{x^{2} - 4 x^{3} < 3 x^{2} - 4 x^{3} > - 3

Solve the inequality for x

More Steps

Evaluate

x^{2} - 4 x^{3} < 3

Move the expression to the left side

x^{2} - 4 x^{3} - 3 < 0

Rewrite the expression

x^{2} - 4 x^{3} - 3 = 0

Find the critical values by solving the corresponding equation

x \approx - 0.832406

Determine the test intervals using the critical values

x < - 0.832406 x > - 0.832406

Choose a value form each interval

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

To determine if x < - 0.832406 is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

Evaluate

(- 2)^{2} - 4 (- 2)^{3} - 3 < 0

Simplify

33 < 0

Check the inequality

false

x < - 0.832406 is not a solution x_{2} = 0

To determine if x > - 0.832406 is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

More Steps

Evaluate

0^{2} - 4 \times 0^{3} - 3 < 0

Simplify

- 3 < 0

Check the inequality

true

x < - 0.832406 is not a solution x > - 0.832406 is the solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x > - 0.832406

x > - 0.832406

{x > - 0.832406 x^{2} - 4 x^{3} > - 3

Solve the inequality for x

More Steps

Evaluate

x^{2} - 4 x^{3} > - 3

Move the expression to the left side

x^{2} - 4 x^{3} - (- 3) > 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x^{2} - 4 x^{3} + 3 > 0

Factor the expression

(- x + 1) (3 x + 4 x^{2} + 3) > 0

Separate the inequality into 2 possible cases

{- x + 1 > 0 3 x + 4 x^{2} + 3 > 0 {- x + 1 < 0 3 x + 4 x^{2} + 3 < 0

Solve the inequality

More Steps

Evaluate

- x + 1 > 0

Move the constant to the right side

- x > 0 - 1

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

- x > - 1

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

x < 1

{x < 1 3 x + 4 x^{2} + 3 > 0 {- x + 1 < 0 3 x + 4 x^{2} + 3 < 0

Solve the inequality

More Steps

Evaluate

3 x + 4 x^{2} + 3 > 0

Move the constant to the right side

3 x + 4 x^{2} > 0 - 3

Add the terms

3 x + 4 x^{2} > - 3

Evaluate

x^{2} + \frac{3}{4} x > - \frac{3}{4}

Add the same value to both sides

x^{2} + \frac{3}{4} x + \frac{9}{64} > - \frac{3}{4} + \frac{9}{64}

Evaluate

x^{2} + \frac{3}{4} x + \frac{9}{64} > - \frac{39}{64}

Evaluate

(x + \frac{3}{8})^{2} > - \frac{39}{64}

Calculate

x \in R

{x < 1 x \in R {- x + 1 < 0 3 x + 4 x^{2} + 3 < 0

Solve the inequality

More Steps

Evaluate

- x + 1 < 0

Move the constant to the right side

- x < 0 - 1

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

- x < - 1

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

x > 1

{x < 1 x \in R {x > 1 3 x + 4 x^{2} + 3 < 0

Solve the inequality

More Steps

Evaluate

3 x + 4 x^{2} + 3 < 0

Move the constant to the right side

3 x + 4 x^{2} < 0 - 3

Add the terms

3 x + 4 x^{2} < - 3

Evaluate

x^{2} + \frac{3}{4} x < - \frac{3}{4}

Add the same value to both sides

x^{2} + \frac{3}{4} x + \frac{9}{64} < - \frac{3}{4} + \frac{9}{64}

Evaluate

x^{2} + \frac{3}{4} x + \frac{9}{64} < - \frac{39}{64}

Evaluate

(x + \frac{3}{8})^{2} < - \frac{39}{64}

Calculate

x \in / R

{x < 1 x \in R {x > 1 x \in / R

Find the intersection

x < 1 {x > 1 x \in / R

Find the intersection

x < 1 x \in / R

Find the union

x < 1

{x > - 0.832406 x < 1

Solution

- 0.832406 < x < 1

Alternative Form

x \in (- 0.832406, 1)

Show Solution