1.6) Linear Inequalities MA 180 Lecture Chapter 1 College Algebra and Calculus by Larson/Hodgkins Equations and Inequalities Simple inequalities are used to order real numbers. To solve an inequality in the variable x we find all values of x for which the inequality is true. Such values are called solutions and are said to satisfy the inequality. The set of all real numbers that are the solutions of an inequality is the solution set of the inequality. The set of points on the real number line that represent the solution set of an inquality is the graph of the inequality. There are four different types of bounded intervals. Let a and b be real numbers such that a<b. The following intervals of the real number line are bounded. The numbers a and b are the endpoints of each interval. Notation Interval Type Inequality Graph Closed a x b Open a x b a x b a x b Note that a closed interval contains both of its endpoints and an open interval does not contain either of its endpoints. Often, the solution of an inequality is an interval on the real line that is unbounded. For instance, the interval consisting of all positive numbers is unbounded. The symbols, positive infinity, and, negative infinity, do not represent real numbers. Let a and b be real numbers. The following intervals of the real number line are unbounded.. Notation Interval Type Inequality Graph a, x a a, Open x a, b Open x b,b x b, Entire real line x Example: Write an inequality to represent each of the following intervals. Then state whther the interval is bounded or unbounded. 5, 2,7
3, 6 Properties of Inequalities We approach this in a way similar to solving linear equations. We want to isolate the variable and can use properties of inequalities to do this. Most properties are the same as in equalities, except we must remember that if we should multiply or divide by a negative number we must switch the inequality sign. Two inequalities that have the same solutions set are equivalent. Let a,b,c, and d be real numbers 1. Transitive Property If a b and b c then a c 2. Addition of Inequalities If a b and c d then a c b d 3. Addition of a Constant If a b then a c b c 4. Multiplication by a Constant For c 0, if a b then ac bc For c 0, if a b then ac bc These properties hold with either a strict inequality (<) or just inequalities. Solving a Linear Inequality We solve this exactly as we would solve a regular linear equation with the one exception, if we should multiply of divide by a negative number we switch the inequality sign. Example: 5 4x 2 2x 8
A double inequality combines two inequalities. For example 2 9 10 is really the same as saying 2 9 and 9 10. We can solve double inequalities by doing the same to all parts. Solve 2 9 10 Inequalities Involving Absolute Value There are two ways to solve inequalities involving absolute values. The first is the way the book describes and involves memorizing two rules. The second method will be similar to what we will use in the next section. We will consider both methods. Let x be a variable or an algebraic expression and let a be a real numbers such that a 0. 1. The solution of x a are all of the values of x that lie between a and a. x a if and only if a x a 2. The solution of x a are all of the values of x that are less than a and greater than a. x a if and only if x a or x a These rules are also valid if < is replaced by and > is replaced by. Use the above rules to solve 2x 3 5 4 1
The second method says to ignore the inequality and replace the inequality sign with an equals sign. Then solve as usual. The resulting solutions are used to divide up the real number line. Then for each subdivision of the number line we will pick a test number. We plug it back into the original inequality and if it holds true we keep that section of the real number line. If it is false we reject that sub-interval. Solve the following using this method. 24x 1 20 12 5 Exercises: Solve 10x 50 2x 7 3 4x
x 1 7 2x 8 3 2 13 8 1 3 x 5x 10 2x 1 2 6 2 x 10 9