Solving Polynomial Inequalities

Notation Review

Above image is closed interval notation meaning the numbers on the line are inclusive. Open interval notation (circle instead of a dot) means the numbers are exclusive. An arrow means the interval extends to infinity.

Symbols	Interval	Meaning
\(a \leq x < b\)	\([a, b)\)	x is greater than or equal to a and less than b
\(x \geq a\)	\([a, +\infty)\)	x is greater than or equal to a
\(x\geq a\text{ or }x\leq b\)	\((-\infty ,b]\cup[a,+\infty)\)	x is less than or equal to b or x is greater than or equal to a

When there are two conditions and both are satisfiable, use an and statement. If there are two conditions and only one is satisfiable at a time, use an or statement. Infinity is always not inclusive because there is no real value for infinity.

Types of Notation

Interval Notation

Uses brackets, ( and ) for not including, [ and ] for including. Eg. (5, 6] would mean x is between 5 and 6, 6 inclusive, 5 exclusive.

Inequality Notation

Uses the \(\leq\), < and \(\geq\), > signs. When there’s a line under the sign, it’s inclusive, otherwise, exclusive. The sign’s opening will always point towardsthe larger number. Eg. \(5 < x \leq 6\) would be the same as the statement above.

Set Notation

Pretty much the same as inequality notation except within a set. Eg. \({x| 5 < x \leq 6}\) means \(x\) such that \(x\) belongs to this set of numbers (in this case between 5 and 6,exlcusive of 5, inclusive of 6).

Number Line Notation

Number lines show an open circle for exclusive, a closed circle for inclusive, and an arrow to extend to \(\pm\infty\)

Solving

General solving method is to find all the values where the function is equal to zero, then find the intervals where the condition is satisfied. Some ways to do this include tables (slow), graphing, and number line (faster).

Assuming \(f(x)\) is in red, the blue part shows \(f(x) > 0\)