Unit 4.5 Study Notes (Carnegie book 7.1) Solving Polynomial Inequalities
Linear Inequalities (the highest power on an 𝒙 is 1): Solving linear inequalities is almost the same as solving linear
equations. To solve a linear inequality you just isolate 𝑥. Just remember that have to switch the order of the
inequality sign if you multiply or divide both sides of the inequality by a negative number.
Interval Notation: Interval notation is a common notation for writing solutions to inequalities. Interval notation uses
parentheses ( , ) and brackets [ , ] . ( , ) are said to be open ( 𝑑𝑜 𝑛𝑜𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝑡ℎ𝑒 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 ) and [ , ] are
said to be closed [ 𝑖𝑛𝑐𝑙𝑢𝑑𝑒𝑠 𝑡ℎ𝑒 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 ]. An interval can be half open and half closed: ( , ] or [ , ). If the
solutions to an inequality go forever to the left or forever to the right, then parenthesis are used.
Example 1: Find the solution set for 𝟓𝒙 − 𝟕 > 𝟑𝒙 + 𝟗. Solve the inequality, graph your solution on a number line,
and write your final answer in interval notation.
Example 2. Graph the inequalities on a number line and then rewrite the solution in interval notation.
A. 2 < 𝑥 < 5
B. −2 ≤ 𝑥 ≤ 0
C. 𝑥 < 5
D. 𝑥 ≥ 0
Note: Always use parenthesis, ( or ) for > or < and use brackets, [ or ] for ≥ or ≤ . Remember that −∞ and ∞ are
never bracketed.
Example 3: Solve 𝟏 − 𝟐𝒙 ≥ 𝒙 − 𝟒 algebraically and express the solution set in
interval notation.
Remember if you multiply
or divide both sides of an
inequality by a negative
number you must switch the
order of the inequality sign!
Example 4: Solve −𝟑 ≤ 𝟔𝒙 − 𝟏 < 𝟑 , graph the solution on a number line, and write the answer in interval notation.
Example 5: Solve −𝟐(𝒙 − 𝟑) > −𝟒 , graph the solution on a number line. Write answer in interval notation.
Solving Quadratic and Higher Order Polynomials (any power greater than 1 on x) is more complicated. One way to
solve is to look at the graph of the polynomial.
When the polynomial is greater than 0, we look for x values that produce a graph ABOVE the x axis.
When the polynomial is less than 0, we look for x values that produce a graph BELOW the x axis.
Example 6: Solve each of the following inequalities by looking at the graph (use the zeros and end behavior). Write
solutions in interval notation.
A. (𝒙 − 𝟑)(𝒙 + 𝟐) < 𝟎
B. 𝒙𝟐 + 𝟓𝒙 ≥ 𝟏𝟒
C. 𝒙𝟐 + 𝟒𝒙 + 𝟒 > 𝟎
E.
𝒙𝟐 + 𝟒𝒙 + 𝟒 ≤ 𝟎
G. Solve 𝒙𝟑 + 𝟓𝒙𝟐 − 𝟓𝒙 − 𝟐𝟓 ≥ 𝟎 by graphing.
Write solution in interval notation.
D. 𝒙𝟐 + 𝟒𝒙 + 𝟒 < 𝟎
F. 𝟐𝒙𝟒 − 𝟖𝒙𝟐 > 𝟎