Question

Solve the inequality algebraically. 2x3 > - 18x2 List the intervals and sign in each interval. Complete the following table. (Type your answers in interval notation. Use ascending order.) Interval Sign What is the solution? (Type your answer in interval notation. Simplify your answer. Use integers or fractions (x - 5)(x-6)(x-7) s0 List the intervals and sign in each interval. Complete the following table. (Type your answers in interval notation. Use ascending order.) Interval Sign What is the solution? (Type your answer in interval notation Simplify your answer Use integers or fractions for any (X-6)2 20 X2 - 1 List the intervals and sign in each interval. Complete the following table. (Type your answers in interval notation. Use ascending order) Interval Sign Negative Positive Negative Positive What is the solution? (-11)4[6,6) (Simplify your answer Type your answer in interval notation Use integers or fractions for any

Answer 1

To solve the inequality 2x^3 > -18x^2, we factor the expression and use a number line to determine the sign in each interval. The solution is x ∈ (-∞, -9) ∪ (0, ∞).

Step-by-step explanation:

To solve the inequality 2x^3 > -18x^2, we need to find the values of x that satisfy the inequality. Here are the steps:

First, rewrite the inequality as 2x^3 + 18x^2 > 0.

Next, factor out the common term x^2 from both terms: x^2(2x + 18) > 0.

Set each factor equal to zero and solve for x to find the critical points: x^2 = 0 and 2x + 18 = 0. The solution for x^2 = 0 is x = 0, and the solution for 2x + 18 = 0 is x = -9.

Create a number line with these critical points and test intervals to determine the sign of the expression. The intervals are (-∞, -9), (-9, 0), and (0, ∞).

Choose a test point from each interval and substitute it into the inequality to determine the sign in each interval. For example, if we choose x = -10, then 2(-10)^3 + 18(-10)^2 = -1820, which is negative. We can do the same for other test points.

Based on the signs, we can conclude that the solution to the inequality is x ∈ (-∞, -9) ∪ (0, ∞) in interval notation.

Answer 2

To solve the inequality 2x^3 > -18x^2, divide both sides by 2x^2 to get x > -9, considering x is not zero. The solution to the inequality is the interval (-9, ∞), as values within this range satisfy the inequality.

Step-by-step explanation:

To solve the inequality 2x^3 > -18x^2 algebraically, we must first simplify the inequality by dividing both sides by 2x^2, assuming x is not zero. The simplified inequality is x > -9. Now, we will consider the sign of the inequality within different intervals determined by the critical point, which is x = -9.

The intervals are as follows:

Interval (-∞, -9): Negative

Interval (-9, ∞): Positive

The solution to the inequality is the set of all x values that satisfy the inequality, which is the interval (-9, ∞).