PreCalculus I

1.7 – Inequalities

Video 1 of 3: Linear Inequalities

Video 2 of 3: Quotient Inequalities

Video 3 of 3: Temperature

Section Notes

Rules for Inequalities

$\begin{array}{|l|l|p{10cm}|} \hline & \text{\textbf{Rule}} & \textbf{Description} \\ \hline\textbf{1.}& \text{If } A\le B \text{ then } A+c \le B+c & Adding the same number to both sides of an inequality yields an equivalent inequality.\\ \hline\textbf{2.}& \text{If } A\le B \text{ then } A-c \le B-c & Subtracting the same number to both sides of an inequality yields an equivalent inequality.\\ \hline\textbf{3.}& \text{If } c>0 \text{ and } A \le B \text{ then } cA\le cB & Multiplying each side of an inequality by the same \emph{positive} value yields an equivalent inequality.\\ \hline\textbf{4.}& \text{If } c<0 \text{ and } A \le B \text{ then } cA\ge cB & Multiplying each side of an inequality by the same \emph{negative} value \underline{\color{red}reverses the direction} of the inequality.\\ \hline\textbf{5.}& \text{If } A>0, B>0, \text{ and } A\le B, \text{ then }\frac{1}{A}\ge\frac{1}{B}& Taking reciprocals of each side of an inequality involving positive quantities \underline{\color{red}reverses the direction} of the inequality.\\ \hline\textbf{6.}& \text{If } A\le B, \text{ and } C\le D, \text{ then } A+C\le B+D & Inequalities can be added \emph{if their signs are in the same direction}.\\ \hline\end{array}$

Sign of Product or Quotient

If a product or quotient has an even number of negative factors, then the product is positive.

If a product or quotient has an odd number of negative factors, then the product is negative.

Solving Nonlinear Inequalities

$\begin{array}{|l|p{12cm}|} \hline\textbf{1.} & \textbf{Move all terms} to one side. If denominators are present, ensure all terms have the common denominator.\\ \hline\textbf{2.} & \textbf{Factor.} Factor the nonzero side of the inequality completely.\\ \hline\textbf{3.} & \textbf{Find the intervals.} By first determining where the nonzero side of the inequality is equal to zero, or is undefined. Then list all the intervals determined by these numbers.\\ \hline\textbf{4.} & \textbf{Make a diagram.} Make a diagram of the number line cut into the intervals found above. \\ \hline\textbf{5.} & \textbf{Solve.} Determine the sign of the nonzero expression for each interval.\\ \hline\textbf{6.} & \textbf{Interpret.} State your answer as an interval, or graphically\\ \hline\end{array}$