How do you solve the polynomial inequality and state the answer in interval notation given #x^6+x^3>=6#?
The inequality is Quadratic in form.
Step 1: We require zero on one side.
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.
If it is more helpful, you may substitute
Step 3: Set each factor equal to zero separately, and solve the equation
These are the two real roots of the equation.
They separate the real line into three intervals:
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
and positive on
Step 5: Complete the problem.
We are interested in knowing where
We know now where the left side equals 0, and we know where it is positive. Write this information in interval form as:
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