Download "and" inequalities

6-4A Solving Compound
Inequalities Involving “AND”
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
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
intersection (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
n2
Write the two inequalities.
2n 8
Rewrite as a single
inequality with the
O
8
2
variable in the middle.
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.
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.
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 weight restriction
and 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.
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 intersection (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 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: An inequality sign points toward the smaller value –
thus both inequality signs point in the same direction - 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
Solve. Then graph the solution.
Example 6
 5  2x  3  7
Example 7
 3  1  2x  5
Example 8
 2  2x  1  7
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.
6-A5 Handout A5 (Study Guide and Intervention Page 27).