Solve -4z^31 \geq 17z^23

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

z \in (- \infty, - \frac{51}{4}] \cup {0}

Evaluate

- 4 z^{3} \times 1 \geq 17 z^{2} \times 3

Multiply the terms

- 4 z^{3} \geq 17 z^{2} \times 3

Multiply the terms

- 4 z^{3} \geq 51 z^{2}

Move the expression to the left side

- 4 z^{3} - 51 z^{2} \geq 0

Rewrite the expression

- 4 z^{3} - 51 z^{2} = 0

Factor the expression

- z^{2} (4 z + 51) = 0

Divide both sides

z^{2} (4 z + 51) = 0

Separate the equation into 2 possible cases

z^{2} = 0 4 z + 51 = 0

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

z = 0 4 z + 51 = 0

Solve the equation

More Steps

z = 0 z = - \frac{51}{4}

Determine the test intervals using the critical values

z < - \frac{51}{4} - \frac{51}{4} < z < 0 z > 0

Choose a value form each interval

z_{1} = - 14 z_{2} = - 6 z_{3} = 1

To determine if z < - \frac{51}{4} is the solution to the inequality,test if the chosen value z = - 14 satisfies the initial inequality

More Steps

z < - \frac{51}{4} is the solution z_{2} = - 6 z_{3} = 1

To determine if - \frac{51}{4} < z < 0 is the solution to the inequality,test if the chosen value z = - 6 satisfies the initial inequality

More Steps

z < - \frac{51}{4} is the solution - \frac{51}{4} < z < 0 is not a solution z_{3} = 1

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

More Steps

z < - \frac{51}{4} is the solution - \frac{51}{4} < z < 0 is not a solution z > 0 is not a solution

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

z \leq - \frac{51}{4} is the solution z = 0

Solution

z \in (- \infty, - \frac{51}{4}] \cup {0}

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