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

##### 1 Answer

The inequality is Quadratic in form.

#### Explanation:

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.

Just as

We treat

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

positive on

negative on

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