Intermediate Algebra 098AChapter 9

Inequalities and Absolute Value

• Albert Einstein

»“In the middle of difficulty lies opportunity.”

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.

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

Example solve by graphing

15 1

15 1

x x

x x

Addition Property of Inequality

• If a < b, then a + c = b + c

• for all real numbers a, b, and c

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

Compound Inequalities 9.1

• Def: Compound Inequality: Two inequalities joined by “and” or “or”

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.

A B

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.

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.

A B

Solving inequalities involving “or”

• Solve each inequality in the compound inequality

• The solution set will be the union of the individual solution sets.

Confucius

–“It is better to light one small candle than to curse the darkness.”

Intermediate Algebra 098A

• Section 9.2

• Absolute Value Equations

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.

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

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

Intermediate Algebra 098A

• Section 9.3

• Absolute Value Inequalities

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.

Sample Problem

• |5x +1| + 1 < 10

• Answer [-2, 8/5]

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.

Sample Problem

8 5 3 11x

Answer

6( ,0] [ , )

5

Intermediate Algebra 9.4

Graphing Linear Inequalities in Two

Variables and Systems of Linear Inequalities

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 >

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.

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.

Joe Namath - quarterback

•“What I do is prepare myself until I know I can do what I have to do.”

Linear inequalities on calculator

• Set =• Solve for Y• Input in Y=• Scroll left and scroll through icons

and press [ENTER]• Press [GRAPH]

Calculator Problem

42

5y x

Abraham Lincoln U.S. President

•“Nothing valuable can be lost by taking time.”