Transcript compound inequality

3.4A Solving Compound
Inequalities
For this lesson you will need a:
What is a compound sentence in your English class?
This weekend I will go shopping and I will go to a
movie.
This weekend I will go shopping or I will go to a
movie.
A compound inequality consists of two inequalities
connected by the word “AND” or the word “OR”.
Today we will work with the type connected by the word
“AND”.
Compound inequalities involving “AND” consist of two
inequalities. Both inequalities must be satisfied to make
the compound inequality true.
If you want to go to the movies with your friends
you must finish your homework and clean your room.
Writing Compound Inequalities with “AND”.
Write a compound inequality that represents the set of
all real numbers greater than or equal to 0 and less than 4.
Then graph the inequality.
n4
and
n0
Compound inequalities with
0 n  4
“AND” are generally
O
combined in a single
4
0
inequality with the
variable in the middle.
•
Both inequalities must be satisfied to make the compound
inequality true. Therefore the solution is the overlap of
the two rays. The graph is a line segment! The number
of solutions are limited.
Write a compound inequality that represents the set of
all real numbers greater than or equal to 2 and less than 8.
Then graph the inequality.
n8
and
1. Write the two inequalities. n  2
2n 8
2. Rewrite as a single
inequality with the
O
8
2
variable in the middle.
3. Graph.
•
Note: A compound inequality is usually written in a
way to reflect the order of numbers on a number line.
Example 1 Write a compound inequality that represents
the set of all real numbers greater than or equal to -12
and less than -5. Then graph the inequality.
n  5
n  12 and
 12  n  5
•
-12
O
-5
Note:
A compound
inequality
is usually
written
in a
In a compound
inequality
involving
AND, both
inequalities
way
the
of numbers
a numbertrue.
line.
mustto
bereflect
satisfied
toorder
make the
compoundoninequality
Therefore the solution is the overlap of the two rays.
The graph is a line segment! The number of solutions are
limited.
Example 2 Write a compound inequality that represents
the set of all real numbers greater than –2 and less than
or equal to 3. Then graph the inequality.
n  2 and
n 3
2  n  3
O
–2
•
3
You must
label the
endpoints!
Example 3 Magic Mountains newest roller coaster ride has
weight restrictions of greater than or equal to 50 pounds
and less than 250 pounds.Write a compound inequality to
describe the the weight restriction and then graph the
inequality.
w  50 and w  250
50  w  250
•
50
O
250
Example 4 Write a compound inequality to describe
when water is liquid the temperature is greater than 32
degrees F and less than 212 degrees F. Then graph the
inequality.
Example 4 Write a compound inequality to describe
when water is liquid the temperature is greater than 32
degrees F and less than 212 degrees F. Then graph the
inequality.
d  32 and d  212
32  d  212
O
32
O
212
Write an inequality that describes the graph.
8  x  3
•
–8
O
3
1. Identify the endpoints.
2. Name the variable.
3. Write the inequality sign
for each endpoint.
Note: The inequality sign points toward the smaller value
– thus the inequality sign points to the left.
Example 5 Write an inequality that describes the graph.
 14  x   2
•
–14
•
–2
1. Identify the endpoints.
2. Name the variable.
3. Write the inequality sign for each endpoint.
Solving Compound Inequalities with “AND”.
To solve compound inequalities with “and”, isolate the
variable. You must perform the operation on all three
expressions.
2  x 2  4
2
2 2
4x2
•
–4
O
2
Example 6 Solve – 5 < 2x + 3 < 7. Then graph the solution.
 5  2x  3  7
3
3 3
 8  2x  4
2 2 2
4x2
•
–4
O
2
Example 7 Solve –3 < –1 – 2x < 5. Then graph the solution.
Reverse both
inequality symbols.
 3  1  2x  5
1
1 1
 2 / 2x / 6
2 > 2 > 2
1  x  3
3  x  1
•
–3
O
1
Example 8 Solve –2 < –2x + 1 < 7. Then graph the solution.
 2  2x  1  7
1
1 1
3 
Reverse both
/ 2x / 6
inequality symbols.  2 >  2 >  2
3
 x  3
2
3
3  x 
2
•
–3
O
3
2
To solve a compound inequality involving “AND”,
isolate the variable (in the center).
To perform any operation on a compound
inequality involving “AND”, you must perform
the operation on all three expressions.
The graph of the solutions to a compound
inequality involving “AND” is a line segment.
A compound inequality consists of two inequalities
connected by the word “AND” or the word “OR”.
Compound inequalities involving
“AND” consist of two inequalities.
Both must be satisfied to make
the compound inequality true. The
solution of these two inequalities will
graph as a line segment.
If you want to go to the movies with your friends
you must finish your homework and clean your room.
If you want to go to the movies with your friends
you must finish your homework or clean your room.
Compound inequalities involving “OR” consist of two
inequalities, either of which can be satisfied to make the
compound inequality true. The solution of these two
inequalities will graph as opposite rays.
O R
Opposite Rays
Compound inequalities with “OR” are written as two
inequalities.
I need to
write all of
the above in
my notes!
Writing Compound Inequalities with
“OR”
Write a compound inequality that represents the set of all
real numbers less than –3 or greater than 0. Then graph
the inequality.
n  3 or n  0
O
–3
O
0
Compound inequalities involving OR will graph as opposite
rays. Notice the inequality symbols are pointing away from
each other. There are infinite solutions!
Write a compound inequality that represents the set of all
real numbers greater than 0 or less than –3. Then graph
the inequality.
or
n0
or nn  3
Reposition the
inequalities to
reflect the
O
O
endpoints order
–3
0
on a number line.
More notes
to record!
Example 1 Soup Plantation offers a reduced price for
children less than or equal to 12 years of age or for adults
over age 65. Write a compound inequality to describe the
age of those people who can eat for a reduced price. Then
graph the inequality.
1. Write the two inequalities.
2. Graph.
a  12
•
12
or
a  65
O
65
Solving Compound Inequalities with “OR”.
Write both inequalities.
Solve each inequality
independently.
Write “or” between
the inequalities.
x4  3
4 4
x  7
•
7
or
2x  18
2
2
x  9
O
9
Compound inequalities involving OR will graph as opposite
rays. Notice the inequality symbols are pointing away from
each other. There are infinite solutions!
Example 2 Solve the inequality. Then graph the solution.
x5  6
5
5
x   11
O
–11
or
or
3x  12
3
3
•
4
x4
Example 3 Solve the inequality. Then graph the solution.
Reposition the
inequalities to
reflect the
endpoints order
on a number line.
8x  1  25 or 6x  5  7
1
1
5 5
6x  12
8x  24
6
6
8
8
x  2
x
x 3
or
O
2
O
3
Example 4 Solve the inequality. Then graph the solution.
 32y  64  32 or  32y  64   32



64
 64
 64 64
Reverse sign
 32y /  32
 32y /  96
when multiplying
 32 <  32
 32 >  32
by a negative.
y 3
or
y  1
O
O
1
3
Example 5 Tell whether –5 is a solution to the compound
inequality.
x  5
or
x  4
1. Write the inequalities.
 5 /  5
or
5 
2. Substitute.
/ 4
false
3. Determine whether the
false
inequalities are true.
No solution
4. Solution if at least one
of the inequalities is true.
Example 6 Tell whether –5 is a solution to the compound
inequality.
x  3
or
x  0
1. Write the inequalities.
5  3
or
5 
/ 0
2. Substitute.
3 Determine whether the
inequalities are true.
4. Solution if at least one
of the inequalities is true.
false
true
solution
The graph of the solution to a
compound inequality involving “AND”
is a line segment.
The graph of the solution to a
compound inequality involving “OR” is
two rays going in opposite direction
(opposite rays).