intermediate algebra 098a chapter 9 inequalities and absolute value
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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.”