Solve g*g^23-1≤2

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

Solve for g

g \leq 1

Alternative Form

g \in (- \infty, 1]

Evaluate

g \times g^{2} \times 3 - 1 \leq 2

Multiply

More Steps

Evaluate

g \times g^{2} \times 3

Multiply the terms with the same base by adding their exponents

g^{1 + 2} \times 3

Add the numbers

g^{3} \times 3

Use the commutative property to reorder the terms

3 g^{3}

3 g^{3} - 1 \leq 2

Move the expression to the left side

3 g^{3} - 1 - 2 \leq 0

Subtract the numbers

3 g^{3} - 3 \leq 0

Rewrite the expression

3 g^{3} - 3 = 0

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

3 g^{3} = 0 + 3

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

3 g^{3} = 3

Divide both sides

\frac{3 g ^{3}}{3} = \frac{3}{3}

Divide the numbers

g^{3} = \frac{3}{3}

Divide the numbers

More Steps

Evaluate

\frac{3}{3}

Reduce the numbers

\frac{1}{1}

Calculate

1

g^{3} = 1

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

3 g^{3} = 31

Calculate

g = 31

Simplify the root

g = 1

Determine the test intervals using the critical values

g < 1 g > 1

Choose a value form each interval

g_{1} = 0 g_{2} = 2

To determine if g < 1 is the solution to the inequality,test if the chosen value g = 0 satisfies the initial inequality

More Steps

Evaluate

3 \times 0^{3} - 1 \leq 2

Simplify

More Steps

Evaluate

3 \times 0^{3} - 1

Calculate

3 \times 0 - 1

Any expression multiplied by 0 equals 0

0 - 1

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

- 1

- 1 \leq 2

Check the inequality

true

g < 1 is the solution g_{2} = 2

To determine if g > 1 is the solution to the inequality,test if the chosen value g = 2 satisfies the initial inequality

More Steps

Evaluate

3 \times 2^{3} - 1 \leq 2

Simplify

More Steps

Evaluate

3 \times 2^{3} - 1

Multiply the terms

24 - 1

Subtract the numbers

23

23 \leq 2

Check the inequality

false

g < 1 is the solution g > 1 is not a solution

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

g \leq 1 is the solution

Solution

g \leq 1

Alternative Form

g \in (- \infty, 1]

Show Solution