Download Intermediate Algebra 098A

Intermediate Algebra 098A
Chapter 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  x  x  1
15  x  x  1
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 x  3  11
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 set
of 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
4
y
x2
5
Abraham Lincoln U.S. President
•“Nothing valuable
can be lost by
taking time.”