MTH 161 Ch2 Polynomial and Rational Inequalities (2.7)

Steps to solve a polynomial inequality?

Move everything to one side so it is compared to 0, factor, find the zeros, mark them on a number line, test a point in each interval, and write the solution in interval notation.

Why does the sign only change at zeros?

A polynomial can only switch between positive and negative where it equals zero (or is undefined for rational functions).

Solve x^2 - x - 6 >= 0.

Factor (x-3)(x+2) >= 0. Zeros -2 and 3. Positive outside the zeros: (-infinity, -2] union [3, infinity).

Solve x^2 - 4 < 0.

(x-2)(x+2) < 0. Negative between the zeros: (-2, 2).

Brackets or parentheses for the endpoints?

Less than or greater than: parentheses. Less than or equal or greater than or equal: brackets (for polynomials).

Steps to solve a rational inequality?

Move everything to one side, combine into a single fraction, factor top and bottom, find zeros of the numerator and denominator, test each interval.

Can you multiply both sides by the denominator in a rational inequality?

No. You do not know its sign, so it might flip the inequality. Get zero on one side instead.

What are the critical values of a rational inequality?

The zeros of the numerator and the zeros of the denominator.

Denominator zeros in the final answer?

Always excluded with a parenthesis, even if the inequality includes equal to.

Solve (x + 1) / (x - 2) > 0.

Critical values -1 and 2. Test the intervals: positive for x less than -1 and x greater than 2. Answer: (-infinity, -1) union (2, infinity).

Solve (x + 1) / (x - 2) <= 0.

The fraction is negative between -1 and 2. Include -1 (numerator zero) but not 2 (denominator zero): [-1, 2).

What does a double root do to the sign?

Nothing: an even multiplicity zero does not change the sign, so the graph bounces off the axis.

How does a graph show an inequality?

f(x) > 0 is where the graph is above the x-axis. f(x) < 0 is where it is below.

How do you use a test point?

Pick any easy number in the interval, plug it into the factored form, and look only at the sign of the result.

Common mistake with inequalities?

Forgetting to move everything to one side first, and including a denominator zero in the answer.

How do you write two separate intervals together?

With the union symbol, for example (-infinity, -2] union [3, infinity).