Intermediate Algebra 098A: Inequalities and Absolute Value

1. Intermediate Algebra 098AChapter 9 Inequalities and Absolute Value

2. Albert Einstein • “In the middle of difficulty lies opportunity.”

3. Linear Inequalities – 3.2 • Def: A linear inequality in one variable is an inequality that can be written in the form ax + b < 0 where a and b are real numbers and a is not equal to 0.

4. Solve by Graphing • Graph the left and right sides and find the point of intersection • Determine where x values are above and below. • Solution is x values – y is not critical

5. Example solve by graphing

6. Addition Property of Inequality • If a < b, then a + c = b + c • for all real numbers a, b, and c

7. Multiplication Property of Inequality • For all real numbers a,b, and c • If a < b and c > 0, then ac < bc • If a < b and c < 0, then ac > bc

8. Compound Inequalities 9.1 • Def: Compound Inequality: Two inequalities joined by “and” or “or”

9. Intersection - Disjunction • Intersection: For two sets A and B, the intersection of A and B, is a set containing only elements that are in both A and B.

10. Solving inequalities involving and • 1. Solve each inequality in the compound inequality • 2. The solution set will be the intersection of the individual solution sets.

11. Union - conjunction • For two sets A and B, the union of A and B is a set containing every element in A or in B.

12. Solving inequalities involving “or” • Solve each inequality in the compound inequality • The solution set will be the union of the individual solution sets.

13. Confucius • “It is better to light one small candle than to curse the darkness.”

14. Intermediate Algebra 098A • Section 9.2 • Absolute Value Equations

15. Absolute Value Equations • If |x|= a and a > 0, then • x = a or x = -a • If |x| = a and a < 0, the solution set is the empty set.

16. Procedure for Absolute Value equation |ax+b|=c • 1. Isolate the absolute the absolute value. • 2. Set up two equations • ax + b = c • ax + b = -c • 3. Solve both equations • 4. Check solutions

17. Procedure Absolute Value equations: |ax + b| = |cx + d| • 1. Separate into two equations • ax + b = cx + d • ax + b = -(cx + d) • 2. Solve both equations • 3. Check solutions

18. Intermediate Algebra 098A • Section 9.3 • Absolute Value Inequalities

19. Inequalities involving absolute value |x| < a • 1. Isolate the absolute value • 2. Rewrite as two inequalities • x < a and –x < a (or x > -a) • 3. Solve both inequalities • 4. Intersect the two solutions note the use of the word “and” and so note in problem.

20. Sample Problem • |5x +1| + 1 < 10 • Answer [-2, 8/5]

21. Inequalities |x| > a • 1. Isolate the absolute value • 2. Rewrite as two inequalities • x > a or –x > a (or x < -a) • 3. Solve the two inequalities – union the two sets **** Note the use of the word “or” when writing problem.

22. Sample Problem

23. Answer

25. Intermediate Algebra 9.4 Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities

26. Def: Linear Inequality in 2 variables • is an inequality that can be written in the form • ax + by < c where a,b,c are real numbers. • Use < or < or > or >

27. Def: Solution & solution setof linear inequality • Solution of a linear inequality in two variables is a pair of numbers (x,y) that makes the inequality true. • Solution set is the set of all solutions of the inequality.

28. Procedure: graphing linear inequality • 1. Set = and graph • 2. Use dotted line if strict inequality or solid line if weak inequality • 3. Pick point and test for truth –if a solution • 4. Shade the appropriate region.

29. Joe Namath - quarterback • “What I do is prepare myself until I know I can do what I have to do.”

30. Linear inequalities on calculator • Set = • Solve for Y • Input in Y= • Scroll left and scroll through icons and press [ENTER] • Press [GRAPH]

31. Calculator Problem

32. Abraham Lincoln U.S. President • “Nothing valuable can be lost by taking time.”