How do you solve the polynomial inequality and state the answer in interval notation given x^6+x^3>=6?

1 Answer
Mar 12, 2018

The inequality is Quadratic in form.

Explanation:

Step 1: We require zero on one side.
x^6 + x^3 - 6 ge 0
Step 2: Since the left side consists of a constant term, a middle term, and a term whose exponent is exactly double that on the middle term, this equation is quadratic "in form." We either factor it like a quadratic, or we use the Quadratic Formula. In this case we are able to factor.

Just as y^2 + y - 6 = (y + 3)(y - 2), we now have
x^6 + x^3 - 6 = (x^3 + 3)(x^3 - 2).
We treat x^3 as though it were a simple variable, y.
If it is more helpful, you may substitute y = x^3, then solve for y, and finally substitute back into x.

Step 3: Set each factor equal to zero separately, and solve the equation x^6 + x^3 - 6 = 0. We find where the left side equals zero because these values will be the boundaries of our inequality.

x^3 + 3 = 0
x^3 = -3
x = -root(3)3

x^3 -2 = 0
x^3 = -2
x = root(3)2

These are the two real roots of the equation.
They separate the real line into three intervals:
(-oo, -root(3)3); (-root(3)3, root(3)2); and (root(3)2, oo).

Step 4: Determine the sign of the left side of the inequality on each of the above intervals.

Using test points is the usual method. Select a value from each interval, and substitute it for x in the left side of the inequality. We might choose -2, then 0, and then 2.
You will discover that the Left Hand Side is
positive on (-oo, -root(3)3);
negative on (-root(3)3, root(3)2);
and positive on (root(3)2, oo).

Step 5: Complete the problem.
We are interested in knowing where x^6 + x^3 - 6 ge 0.
We know now where the left side equals 0, and we know where it is positive. Write this information in interval form as:

(-oo, -root(3)3] uu [root(3)2, oo).

NOTE: We have the brackets because the two sides of the inequality are equal at those points, and the original problem requires for us to include those values. Had the problem used > instead of ge, we would have used parentheses.