LESSON 6-3
COMPOUND INEQUALITIES
Objective: To solve and graph inequalities
containing “and” or “or”.
Real-World Connection
 Do you like to swim?
 Does your family have a swimming
pool?
 Determining how to keep the water in
a pool chemically balanced is one way
this skill is used in real-world
situations.
Vocabulary
 compound inequality – two inequalities
joined by the word and or the word or
 solution for and inequalities – (Intersection)
any number that makes both inequalities true
 solution for or inequalities – (Union) any
number that makes either inequality true
STEPS for AND problems
1. Write the compound inequality as two
inequalities joined by and.
2. Solve each inequality.
3. Simplify and write the solutions as one
statement.
4. Graph the solution.
Translate the verbal phrase into an
inequality. Then graph the inequality.
a.
All real numbers that are greater than – 2
and less than 3.
Inequality:
–2<x<3
Graph:
b.
All real numbers that are less than 0 or greater than
or equal to 2.
Inequality:
Graph:
x < 0 or x ≥ 2
c.
All real numbers that are less than –1 or greater than
or equal to 4.
Inequality: x < –1 or x ≥ 4
d.
All real numbers that are greater than or equal
To –3 and less than 5.
Inequality: x ≥ –3 and x < 5
= –3 ≤ x < 5
Write and graph a real-world
inequality:
CAMERA CARS
A crane sits on top of a camera
car and faces toward the front.
The crane’s maximum height
and minimum height above the
ground are shown. Write and
graph a compound inequality
that describes the possible
heights of the crane.
Write and graph a real-world
inequality:
SOLUTION
Let h represent the height (in feet) of the
crane. All possible heights are greater than
or equal to 4 feet and less than or equal to 18
feet. So, the inequality is 4 ≤ h ≤ 18.
You try…
 Solve each inequality and graph the solution.
1) -6 < 3x < 15
2) -3 < 2x-1 < 7
You try…
3) 7 < -3a + 1 < 13
 Solution: -4 < n < -2
You try…
4) The books were priced between $3.50 and
$6.00, inclusive.
 Solution: 3.50 < b < 6
To solve OR problems, solve
each inequality separately.
5) Solve –2x + 7 > 3 or 3x – 4 > 5.
You try…
6) Write an inequality that represents all real
numbers that are at most –5 or at least 3. Graph
your solution.
7) Graph 3 < 2m – 1 < 9
Remember, when written like this, it is
an AND problem!
3 < 2m – 1 AND 2m – 1 < 9
Solve each inequality.
Graph the intersection of
2 < m and m < 5.
-5
0
5
Which inequalities describe
the following graph?
o
o
-3
1. y > -3 or y < -1
2. y > -3 and y < -1
3. y ≤ -3 or y ≥ -1
4. y ≥ -3 and y ≤ -1
Answer Now
-2
-1
Which is equivalent to
-3 < y < 5?
1. y > -3 or y < 5
2. y > -3 and y < 5
3. y < -3 or y > 5
4. y < -3 and y > 5
Answer Now
Which is equivalent to
x > -5 and x ≤ 1?
1. -5 < x ≤ 1
2. -5 > x ≥ 1
3. -5 > x ≤ 1
4. -5 < x ≥ 1
Answer Now
Summary
 Re-write the compound inequality into two
problems and solve.
 For and problems, combine the solutions into
one statement.
 For or problems, solve each separately.
 Graph.
Homework:
 Study for quiz Wednesday, 1/15
 Quiz covers chapter 6 sections 1-3
Exit Ticket
 Explain the difference
between “and” vs. “or”
compound inequalities.
 How do you solve a
compound inequality?