Question

Find integral values of x such that (x⁴ - 3x < x + 12 ≤ -3x + frac29/2).

A. -3, -2, -1, 0
B. -2, -1, 0, 1
C. -1, 0, 1, 2
D. 0, 1, 2, 3

Answer 1

To find the integral values of x for the given inequalities, solve each inequality separately and take the intersection of their solutions. The final set will provide the required integral values of x.

Step-by-step explanation:

We are asked to find integral values of x such that x⁴ - 3x < x + 12 ≤ -3x + ½. To find the solution, we need to solve two separate inequalities and then find the intersection of their solutions.

First, let's solve the inequality x⁴ - 3x < x + 12:

1. Subtract x from both sides to get x⁴ - 4x - 12 < 0.
2. Factor the quartic equation if possible or find the roots by graphing or using another method.
3. Determine the intervals where the inequality holds.

Next, solve the inequality x + 12 ≤ -3x + ½:

1. Add 3x to both sides and subtract 12 to get 4x ≤ -¾.
2. Divide by 4 to isolate x and get x ≤ -½
3. Since we want integral solutions, x can be any integer less than or equal to -1.

Finally, find the set of integers that satisfy both conditions by taking the intersection of the two individual sets.