1 Sections 17.1, 17.2, & 17.4 Linear and Absolute Value Inequalities

2 Representing Inequalities Five inequality symbols: Representing linear inequalities in one variable: Number line graphs Set builder notation Interval notation (see next slide)

3 Interval Notation InequalityIntervalGraph a ≤ x ≤ b[a,b] a ≤ x < b[a,b) a < x ≤ b(a,b] a < x < b(a,b) x ≤ a(-∞,a] x < a(-∞,a) x ≥ b[b,∞) x > b(b,∞) INTERVAL NOTATION is also used to describe sets defined by single or double inequalities, as shown in the following table. a b a b a b a b a b a b a b a b

4 (A) Write x ≥ -2 in interval notation and graph. (B) Write in interval notation and graph. (C) Write -4.6 < x ≤ 0.8 in interval notation and graph. (D) Write [-5, 2) as a double inequality and graph. Examples

5 Representing Inequalities Example: “The set of all real numbers, x, less than or equal to three” Graph: Set builder notation: Interval notation:

6 Properties of Inequalities Property 1: The sense (direction) of an inequality is NOT changed when the same number is added or subtracted to both sides of the inequality. Property 2: The sense (direction) of an inequality is NOT changed if both sides of the inequality are multiplied or divided by the same POSITIVE number. Property 3: The sense (direction) of an inequality is reversed if both sides of the inequality are multiplied or divided by the same NEGATIVE number. Property 4: If both sides of the inequality are positive numbers and n is a positive integer, then the inequality formed by taking the nth root or nth power of each side is in the same sense as as the given inequality.

7 Solving Inequalities Solving a linear inequality is similar to solving a basic linear equation, keeping in mind that if you multiply or divide both sides of the inequality by the same negative number, you must reverse the direction of the inequality symbol. Solve:

8 Solving Inequalities Solve:

9 Solving Inequalities with Three Members Solve: Isolate the variable in the middle; apply inverse operations to all three members.

10 Solving Inequalities with Three Members Solve: You can also split the double inequality into a compound inequality joined by the word “and”. If it is an “and” statement (conjunction), the solution must satisfy BOTH inequalities.

11 Solving Inequalities with Three Members Solve:

12 Application Use an inequality to state the domain of the function:

13 Absolute Value Inequalities Recall that the absolute value of a number is the distance between that number and zero on the number line. So since both 5 and -5 are exactly five units from zero on the number line.

14 Absolute Value Inequalities We will rewrite absolute value inequalities as equivalent compound inequalities that do not involve absolute values.

15 Absolute Value Inequality “Less Than” Example:

16 Absolute Value Inequality “Greater Than” Example:

17 Absolute Value Inequalities

18 Absolute Value Inequalities

19 Absolute Value Inequalities ISOLATE the absolute value expression FIRST!

20

21 Consider the following example of an absolute value inequality: A bolt to be used in automobile manufacturing is to be 0.25 inches wide with a tolerance of inch. The manufacturing specs can be written as an absolute value inequality. Absolute Value Inequalities Acceptable WidthOptimal Width Tolerance

22 To express the acceptable width as an interval, we solve for w by first rewriting the inequality as a compound inequality without using absolute value. Since it is a less than (in the form ) we rewrite the absolute value as the double inequality: Absolute Value Inequalities

23 End of Section