Solve t 1 >t^2 =t^3 >t^4

Question

Solve the inequality

t \in \emptyset

Alternative Form

No solution

Evaluate

t 1 > t^{2} = t^{3} > t^{4}

Separate into two inequalities

⎩ ⎨ ⎧ t \times 1 > t^{2} t^{2} = t^{3} t^{3} > t^{4}

Solve the inequality

More Steps

Evaluate

t \times 1 > t^{2}

Any expression multiplied by 1 remains the same

t > t^{2}

Move the expression to the left side

t - t^{2} > 0

Evaluate

t^{2} - t < 0

Add the same value to both sides

t^{2} - t + \frac{1}{4} < \frac{1}{4}

Evaluate

(t - \frac{1}{2})^{2} < \frac{1}{4}

Take the 2 -th root on both sides of the inequality

(t - \frac{1}{2})^{2} < \frac{1}{4}

Calculate

t - \frac{1}{2} < \frac{1}{2}

Separate the inequality into 2 possible cases

{t - \frac{1}{2} < \frac{1}{2} t - \frac{1}{2} > - \frac{1}{2}

Calculate

More Steps

Evaluate

t - \frac{1}{2} < \frac{1}{2}

Move the constant to the right side

t < \frac{1}{2} + \frac{1}{2}

Add the numbers

t < 1

{t < 1 t - \frac{1}{2} > - \frac{1}{2}

Cancel equal terms on both sides of the expression

{t < 1 t > 0

Find the intersection

0 < t < 1

⎩ ⎨ ⎧ 0 < t < 1 t^{2} = t^{3} t^{3} > t^{4}

Solve the inequality

More Steps

Evaluate

t^{2} = t^{3}

Move the expression to the left side

t^{2} - t^{3} = 0

Factor the expression

t^{2} (1 - t) = 0

Separate the equation into 2 possible cases

t^{2} = 0 \cup 1 - t = 0

The only way a power can be 0 is when the base equals 0

t = 0 \cup 1 - t = 0

Solve the equation

More Steps

Evaluate

1 - t = 0

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

- t = 0 - 1

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

- t = - 1

Change the signs on both sides of the equation

t = 1

t = 0 \cup t = 1

⎩ ⎨ ⎧ 0 < t < 1 t = 0 \cup t = 1 t^{3} > t^{4}

Solve the inequality

More Steps

Evaluate

t^{3} > t^{4}

Move the expression to the left side

t^{3} - t^{4} > 0

Factor the expression

t^{3} (1 - t) > 0

Separate the inequality into 2 possible cases

{t^{3} > 0 1 - t > 0 \cup {t^{3} < 0 1 - t < 0

The only way a base raised to an odd power can be greater than 0 is if the base is greater than 0

{t > 0 1 - t > 0 \cup {t^{3} < 0 1 - t < 0

Solve the inequality

More Steps

Evaluate

1 - t > 0

Move the constant to the right side

- t > 0 - 1

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

- t > - 1

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

t < 1

{t > 0 t < 1 \cup {t^{3} < 0 1 - t < 0

The only way a base raised to an odd power can be less than 0 is if the base is less than 0

{t > 0 t < 1 \cup {t < 0 1 - t < 0

Solve the inequality

More Steps

Evaluate

1 - t < 0

Move the constant to the right side

- t < 0 - 1

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

- t < - 1

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

t > 1

{t > 0 t < 1 \cup {t < 0 t > 1

Find the intersection

0 < t < 1 \cup {t < 0 t > 1

Find the intersection

0 < t < 1 \cup t \in \emptyset

Find the union

0 < t < 1

⎩ ⎨ ⎧ 0 < t < 1 t = 0 \cup t = 1 0 < t < 1

Simplify

{0 < t < 1 t = 0 \cup t = 1

Solution

t \in \emptyset

Alternative Form

No solution

Show Solution