Solving Inequalities

1. 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 + 11 5x– 1. • Note. Use a filled-in circle if endpoint is included and an open circle if endpoint is not included in solution set.

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

3. 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.

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

5. 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:

6. 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 < 1 or x > 5. • Two additional misuses of inequality notation are given in the following examples. Do you see why they are incorrect or misleading?

7. 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

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

9. Graphical interpretation of certain inequalities • Let a be a positive real number. The inequality |x| < a has as solution all x whosedistance 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, ). – a 0 a – a a 0

10. 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?).

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