Solve 12a^6≥123

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for a

a \in (- \infty, - \frac{6 656}{2}] \cup [\frac{6 656}{2}, + \infty)

Evaluate

12 a^{6} \geq 123

Move the expression to the left side

12 a^{6} - 123 \geq 0

Rewrite the expression

12 a^{6} - 123 = 0

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

12 a^{6} = 0 + 123

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

12 a^{6} = 123

Divide both sides

\frac{12 a ^{6}}{12} = \frac{123}{12}

Divide the numbers

a^{6} = \frac{123}{12}

Cancel out the common factor 3

a^{6} = \frac{41}{4}

Take the root of both sides of the equation and remember to use both positive and negative roots

a = \pm 6 \frac{41}{4}

Simplify the expression

More Steps

a = \pm \frac{6 656}{2}

Separate the equation into 2 possible cases

a = \frac{6 656}{2} a = - \frac{6 656}{2}

Determine the test intervals using the critical values

a < - \frac{6 656}{2} - \frac{6 656}{2} < a < \frac{6 656}{2} a > \frac{6 656}{2}

Choose a value form each interval

a_{1} = - 2 a_{2} = 0 a_{3} = 2

To determine if a < - \frac{6 656}{2} is the solution to the inequality,test if the chosen value a = - 2 satisfies the initial inequality

More Steps

a < - \frac{6 656}{2} is the solution a_{2} = 0 a_{3} = 2

To determine if - \frac{6 656}{2} < a < \frac{6 656}{2} is the solution to the inequality,test if the chosen value a = 0 satisfies the initial inequality

More Steps

a < - \frac{6 656}{2} is the solution - \frac{6 656}{2} < a < \frac{6 656}{2} is not a solution a_{3} = 2

To determine if a > \frac{6 656}{2} is the solution to the inequality,test if the chosen value a = 2 satisfies the initial inequality

More Steps

a < - \frac{6 656}{2} is the solution - \frac{6 656}{2} < a < \frac{6 656}{2} is not a solution a > \frac{6 656}{2} is the solution

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

a \leq - \frac{6 656}{2} is the solution a \geq \frac{6 656}{2} is the solution

Solution

a \in (- \infty, - \frac{6 656}{2}] \cup [\frac{6 656}{2}, + \infty)

Show Solution