Solve v* v^3≥14 v

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

v \in (- \infty, 0] \cup [314, + \infty)

Evaluate

v \times v^{3} \geq 14 v

Multiply the terms

More Steps

v^{4} \geq 14 v

Move the expression to the left side

v^{4} - 14 v \geq 0

Rewrite the expression

v^{4} - 14 v = 0

Factor the expression

v (v^{3} - 14) = 0

Separate the equation into 2 possible cases

v = 0 v^{3} - 14 = 0

Solve the equation

More Steps

v = 0 v = 314

Determine the test intervals using the critical values

v < 0 0 < v < 314 v > 314

Choose a value form each interval

v_{1} = - 1 v_{2} = 1 v_{3} = 3

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

More Steps

v < 0 is the solution v_{2} = 1 v_{3} = 3

To determine if 0 < v < 314 is the solution to the inequality,test if the chosen value v = 1 satisfies the initial inequality

More Steps

v < 0 is the solution 0 < v < 314 is not a solution v_{3} = 3

To determine if v > 314 is the solution to the inequality,test if the chosen value v = 3 satisfies the initial inequality

More Steps

v < 0 is the solution 0 < v < 314 is not a solution v > 314 is the solution

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

v \leq 0 is the solution v \geq 314 is the solution

Solution

v \in (- \infty, 0] \cup [314, + \infty)

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