Solving Inequalities We can solve inequalities just like equations, with the following exception: Multiplication or division of an inequality by a negative number reverses the direction of the inequality. Problem. Solve 2x x – 1. Note. Use a filled-in circle if endpoint is included and an open circle if endpoint is not included in solution set.

Compound Inequalities Problem. Solve the inequality Solution. The solution is the half-open interval which may be represented graphically as:

Critical Value Method The Critical Value Method is an alternative to the algebraic approach to solving inequalities. The critical values of an inequality are: 1. those values for which either side of the inequality is not defined (such as a denominator equal to 0). 2. those values that are solutions to the equation obtained by replacing the inequality sign with an equal sign. The critical values determine endpoints of intervals on the number line. A given inequality is satisfied at all points in one of these intervals or it is satisfied at none of these points. In order to find out in which intervals the given inequality holds, we may test any point in each interval.

Example for Critical Value Method Solve the inequality. The solution is the half-open interval which may be represented graphically as:

Solving a polynomial inequality using the critical value method Solve (x – 2)(2x + 5)(3 – x) < 0. The critical values are: Graphically, the solution is: The solution consists of two open intervals:

Misuse of inequality notation Students often write something similar to 1 > x > 5. This is incorrect since x cannot simultaneously be less than 1 and greater than 5. What is likely intended is that either x 5. Two additional misuses of inequality notation are given in the following examples. Do you see why they are incorrect or misleading?

Linear and Quadratic Inequalities; We discussed Solving inequalities is like solving equations with one exception Interval notation and number line representation Critical value method Misuse of inequality notation

Absolute Value in Equations Solve |2x – 7| = 11. When removing absolute value brackets, we must always consider two cases.

Graphical interpretation of certain inequalities Let a be a positive real number. The inequality |x| < a has as solution all x whose distance from the origin is less than a. The solution set consists of the interval (–a, a). The inequality |x| > a has as solution all x whose distance from the origin is greater than a. For this inequality, the solution set consists of the two infinite intervals (– , –a) and (a,  ). 0 a – a 0 a

Absolute Value in Inequalities To solve an inequality involving absolute values, first change the inequality to an equality and solve for the critical values. Once you have the critical values, apply the Critical Value Method. Example. Solve |2x – 6| > 4. First, determine that the critical values are 1 and 5 (do you see why?).

Absolute Value in Equations and Inequalities; We discussed Two cases when absolute value brackets are removed Graphical interpretation of |x| a. Critical value method for absolute value inequalities