Question

|x^3-2x^2+12|≤|x^3+2x^2+4|

Answer 1

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Answer 2

To solve the inequality |x^3 - 2x^2 + 12| ≤ |x^3 + 2x^2 + 4|, we'll consider different cases for the value of x. The absolute value of a number is always non-negative, so we can remove the absolute value notation and work with the inequality in its simpler form:

x^3 - 2x^2 + 12 ≤ x^3 + 2x^2 + 4.

Now, let's solve for x:

Step 1: Subtract x^3 from both sides to get rid of the x^3 terms: -2x^2 + 12 ≤ 2x^2 + 4.

Step 2: Move all the x^2 terms to one side: -4x^2 ≤ 4.

Step 3: Divide both sides by -4 (remembering to reverse the inequality when dividing by a negative number): x^2 ≥ -1.

Now, we need to find the values of x that satisfy this inequality. Since x^2 is always non-negative (for real numbers), the inequality x^2 ≥ -1 is true for all real values of x.

Therefore, the solution to the original inequality |x^3 - 2x^2 + 12| ≤ |x^3 + 2x^2 + 4| is all real numbers: (-∞, +∞). In other words, the inequality holds for all values of x.