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

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.