Solve 4x+6>1-x^3(x-1)≤x+5

Question

Solve the inequality

x \in R

Alternative Form

All real solution

Evaluate

4 x + 6 > 1 - x^{3} (x - 1) \leq x + 5

Separate into two inequalities

{4 x + 6 > 1 - x^{3} (x - 1) 1 - x^{3} (x - 1) \leq x + 5

Solve the inequality

More Steps

Evaluate

4 x + 6 > 1 - x^{3} (x - 1)

Move the expression to the left side

4 x + 6 - (1 - x^{3} (x - 1)) > 0

Calculate the sum or difference

More Steps

Evaluate

4 x + 6 - (1 - x^{3} (x - 1))

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

4 x + 6 - 1 + x^{3} (x - 1)

Subtract the numbers

4 x + 5 + x^{3} (x - 1)

4 x + 5 + x^{3} (x - 1) > 0

Calculate

More Steps

Evaluate

x^{3} (x - 1)

Apply the distributive property

x^{3} \times x - x^{3} \times 1

Multiply the terms

x^{4} - x^{3} \times 1

Any expression multiplied by 1 remains the same

x^{4} - x^{3}

4 x + 5 + x^{4} - x^{3} > 0

Rewrite the expression

4 x + 5 + x^{4} - x^{3} = 0

Find the critical values by solving the corresponding equation

x \in / R

There are no key numbers,so choose any value to test,for example x = 0

x = 0

To determine if the solution to the inequality are all real numbers,test if the chosen value satisfies the initial inequality

More Steps

Evaluate

4 \times 0 + 5 + 0^{4} - 0^{3} > 0

Any expression multiplied by 0 equals 0

0 + 5 + 0^{4} - 0^{3} > 0

Simplify

5 > 0

Check the inequality

true

x \in R

{x \in R 1 - x^{3} (x - 1) \leq x + 5

Solve the inequality

More Steps

Evaluate

1 - x^{3} (x - 1) \leq x + 5

Move the expression to the left side

1 - x^{3} (x - 1) - (x + 5) \leq 0

Subtract the terms

More Steps

Evaluate

1 - x^{3} (x - 1) - (x + 5)

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

1 - x^{3} (x - 1) - x - 5

Subtract the numbers

- 4 - x^{3} (x - 1) - x

- 4 - x^{3} (x - 1) - x \leq 0

Calculate

More Steps

Evaluate

- x^{3} (x - 1)

Apply the distributive property

- x^{3} \times x - (- x^{3} \times 1)

Multiply the terms

- x^{4} - (- x^{3} \times 1)

Any expression multiplied by 1 remains the same

- x^{4} - (- x^{3})

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^{4} + x^{3}

- 4 - x^{4} + x^{3} - x \leq 0

Rewrite the expression

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

Find the critical values by solving the corresponding equation

x \in / R

There are no key numbers,so choose any value to test,for example x = 0

x = 0

To determine if the solution to the inequality are all real numbers,test if the chosen value satisfies the initial inequality

More Steps

Evaluate

- 4 - 0^{4} + 0^{3} - 0 \leq 0

Simplify

- 4 \leq 0

Check the inequality

true

x \in R

{x \in R x \in R

Solution

x \in R

Alternative Form

All real solution

Show Solution