Solve S1 > S^2 > S^3 > S^4

Question

Solve the inequality

0 < S < 1

Alternative Form

S \in (0, 1)

Evaluate

S 1 > S^{2} > S^{3} > S^{4}

Separate into two inequalities

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

Solve the inequality

More Steps

Evaluate

S \times 1 > S^{2}

Any expression multiplied by 1 remains the same

S > S^{2}

Move the expression to the left side

S - S^{2} > 0

Evaluate

S^{2} - S < 0

Add the same value to both sides

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

Evaluate

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

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

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

Calculate

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

Separate the inequality into 2 possible cases

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

Calculate

More Steps

Evaluate

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

Move the constant to the right side

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

Add the numbers

S < 1

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

Cancel equal terms on both sides of the expression

{S < 1 S > 0

Find the intersection

0 < S < 1

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

Solve the inequality

More Steps

Evaluate

S^{2} > S^{3}

Move the expression to the left side

S^{2} - S^{3} > 0

Factor the expression

S^{2} (1 - S) > 0

Separate the inequality into 2 possible cases

{S^{2} > 0 1 - S > 0 {S^{2} < 0 1 - S < 0

Solve the inequality

More Steps

Evaluate

S^{2} > 0

Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of S,except when S^{2} = 0

S^{2} = 0

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

S = 0

Exclude the impossible values of S

S \neq = 0

{S \neq = 0 1 - S > 0 {S^{2} < 0 1 - S < 0

Solve the inequality

More Steps

Evaluate

1 - S > 0

Move the constant to the right side

- S > 0 - 1

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

- S > - 1

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

S < 1

{S \neq = 0 S < 1 {S^{2} < 0 1 - S < 0

Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is false for any value of S

{S \neq = 0 S < 1 {S \in / R 1 - S < 0

Solve the inequality

More Steps

Evaluate

1 - S < 0

Move the constant to the right side

- S < 0 - 1

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

- S < - 1

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

S > 1

{S \neq = 0 S < 1 {S \in / R S > 1

Find the intersection

S \in (- \infty, 0) \cup (0, 1) {S \in / R S > 1

Find the intersection

S \in (- \infty, 0) \cup (0, 1) S \in / R

Find the union

S \in (- \infty, 0) \cup (0, 1)

⎩ ⎨ ⎧ 0 < S < 1 S \in (- \infty, 0) \cup (0, 1) S^{3} > S^{4}

Solve the inequality

More Steps

Evaluate

S^{3} > S^{4}

Move the expression to the left side

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

Factor the expression

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

Separate the inequality into 2 possible cases

{S^{3} > 0 1 - S > 0 {S^{3} < 0 1 - S < 0

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

{S > 0 1 - S > 0 {S^{3} < 0 1 - S < 0

Solve the inequality

More Steps

Evaluate

1 - S > 0

Move the constant to the right side

- S > 0 - 1

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

- S > - 1

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

S < 1

{S > 0 S < 1 {S^{3} < 0 1 - S < 0

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

{S > 0 S < 1 {S < 0 1 - S < 0

Solve the inequality

More Steps

Evaluate

1 - S < 0

Move the constant to the right side

- S < 0 - 1

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

- S < - 1

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

S > 1

{S > 0 S < 1 {S < 0 S > 1

Find the intersection

0 < S < 1 {S < 0 S > 1

Find the intersection

0 < S < 1 S \in \emptyset

Find the union

0 < S < 1

⎩ ⎨ ⎧ 0 < S < 1 S \in (- \infty, 0) \cup (0, 1) 0 < S < 1

Simplify

{0 < S < 1 S \in (- \infty, 0) \cup (0, 1)

Solution

0 < S < 1

Alternative Form

S \in (0, 1)

Show Solution