6.4 Solve Compound
Inequalities
•Students will solve
compound
Inequalities.
Lesson 6.4, For use with pages 379-388
Solve the inequality.
1.
8 > x + 10
ANSWER
all real numbers less than –2
2. 6  2x – 4
ANSWER
all real numbers greater than or equal to 5
Lesson 6.4, For use with pages 379-388
3. You estimate you can read at least 8 history text
pages per day. What are the possible numbers of
day it will take you to read at most 118 pages?
ANSWER
at most 15 days
Daily Homework Quiz
For use after Lesson 6.3
1. Solve 3(x + 8) < 9. Graph your solution.
ANSWER
all real numbers less than –5
Daily Homework Quiz
For use after Lesson 6.3
Solve the inequality, if possible.
2.
6k + 1 > 3(2k – 3)
ANSWER
all real numbers because 1 > –9 is true
3. 5(a – 2)< 5a – 14
ANSWER
no solutions because –10 < –14 is false
Daily Homework Quiz
For use after Lesson 6.3
4. A box of cat treats contains at least 50 treats. So
far you have fed your cats 18 treats. If you want the
box of treats to last 8 days, what are the possible
average numbers of treats you can feed the cats per
day?
ANSWER
an average of no more than 4 treats per day.
Venn Diagram
• Add vs. Or
EXAMPLE 1
Write and graph compound inequalities
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: x < 0 or x  2
Graph:
GUIDED PRACTICE
for Example 1
Translate the verbal phrase into an inequality. Then
graph the inequality.
1.
All real numbers that are less than –1 or greater than
or equal to 4
Inequality: x < –1 or x  4
2.
All real numbers that are greater than or equal
To –3 and less than 5
Inequality: x  –3 and x < 5 = –3  x < 5
EXAMPLE 2 Write and graph a real-world compound 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.
EXAMPLE 2 Write and graph a real-world compound 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.
EXAMPLE 3
Solve a compound inequality with and
Solve 2 < x + 5 < 9. Graph your solution.
SOLUTION
Separate the compound inequality into two
inequalities. Then solve each inequality separately.
2 < x + 5 and
x+5<9
Write two inequalities.
2 – 5 < x + 5 – 5 and x + 5 – 5 < 9 – 5 Subtract 5 from each side.
–3 < x and
x<4
Simplify.
The compound inequality can be written as –3 < x < 4.
EXAMPLE 3
Solve a compound inequality with and
ANSWER
The solutions are all real numbers greater than –3 and
less than 4.
Graph:
GUIDED PRACTICE
for Example 2 and 3
Investing
3.
An investor buys shares of a stock and will sell them
if the change c in value from the purchase price of a
share is less than –$3.00 or greater than $4.50. Write
and graph a compound inequality that describes the
changes in value for which the shares will be sold.
ANSWER
c < –3 or c > 4.5
EXAMPLE
3
for Example
2 and 3 with and
Solve a compound
inequality
GUIDED PRACTICE
Solve the inequality. Graph your solution.
4. –7 < x – 5 < 4
–2 < x < 9
ANSWER
9
Graph:
–6
–4
–2
0
2
4
6
8
10
for Example 2 and 3
GUIDED PRACTICE
Solve the inequality. Graph your solution.
5. 10  2y + 4  24
3  y  10
ANSWER
3
Graph:
0
2
4
6
8
10
12
EXAMPLE
3
for Example
2 and 3 with and
Solve a compound
inequality
GUIDED PRACTICE
Solve the inequality. Graph your solution.
6.
–7 < –z – 1 < 3
ANSWER
–4 < z < 6
EXAMPLE 4
Solve a compound inequality with and
Solve –5  –x – 3  2. Graph your solution.
–5  –x – 3  2
Write original inequality.
–5 + 3  –x – 3 + 3  2 + 3
Add 3 to each expression.
–2  –x  5
–1(–2)  –1(–x)  –1(5)
2  x  –5
Simplify.
Multiply each
expression by –1 and
reverse both inequality
symbols.
Simplify.
EXAMPLE 4
Solve a compound inequality with and
–5  x  2
ANSWER
The solutions are all real
numbers greater than or
equal to –5 and less than
or equal to 2.
Rewrite in the form
a  x  b.
EXAMPLE 5
Solve a compound inequality with or
Solve 2x + 3 < 9 or 3x – 6 > 12. Graph your solution.
SOLUTION
Solve the two inequalities separately.
2x + 3 < 9
or
3x – 6 > 12
2x + 3 – 3 < 9 – 3 or 3x – 6 + 6 > 12 + 6
2x < 6
or
3x > 18
Write original
inequality.
Addition or
Subtraction
property of
inequality
Simplify.
EXAMPLE 5
Solve a compound inequality with or
2x < 6
2
2
or
3x
18
>
3
3
x<3
or
x>6
Division property
of inequality
Simplify.
ANSWER
The solutions are all real numbers less than 3 or greater
than 6.
GUIDED PRACTICE
for Examples 4 and 5
Solve the inequality. Graph your solution.
7.
–14 < x – 8 < –1
ANSWER
–6 < x < 7
GUIDED PRACTICE
for Examples 4 and 5
Solve the inequality. Graph your solution.
8.
–1  –5t + 2  4
ANSWER
– 2 t 3
5
5
GUIDED PRACTICE
for Examples 4 and 5
Solve the inequality. Graph your solution.
9. 3h + 1< – 5 or
ANSWER
2h – 5 > 7
h < –2 or h > 6
GUIDED PRACTICE
for Examples 4 and 5
Solve the inequality. Graph your solution.
10.
4c + 1  –3 or
ANSWER
5c – 3 > 17
c  –1 or c > 4
EXAMPLE 6
Solve a multi-step problem
Astronomy
The Mars Exploration Rovers Opportunity and Spirit
are robots that were sent to Mars in 2003 in order to
gather geological data about the planet. The
temperature at the landing sites of the robots can
range from 100°C to 0°C.
• Write a compound inequality that describes the
possible temperatures (in degrees Fahrenheit) at a
landing site.
• Solve the inequality. Then graph your solution.
• Identify three possible temperatures (in degrees
Fahrenheit) at a landing site.
EXAMPLE 6
Solve a multi-step problem
SOLUTION
Let F represent the temperature in degrees Fahrenheit,
and let C represent the temperature in degrees
Celsius. Use the formula C = 5 (F – 32).
9
STEP 1
Write a compound inequality. Because the
temperature at a landing site ranges from –100°C to
0°C, the lowest possible temperature is –100°C, and
the highest possible temperature is 0°C.
–100  C  0
–100 
5 (F – 32)
0
9
Write inequality using C.
Substitute 5 (F – 32) for C.
9
EXAMPLE 6
Solve a multi-step problem
STEP 2
Solve the inequality. Then graph your solution.
–100 
5 (F – 32)
0
9
Write inequality from Step 1.
–180  (F – 32 )  0
Multiply each expression by
–148  F  32
Add 32 to each expression.
9
.
5
EXAMPLE 6
Solve a multi-step problem
STEP 3
Identify three possible temperatures.
The temperature at a landing site is greater than or
equal to –148°F and less than or equal to 32°F. Three
possible temperatures are –115°F, 15°F, and 32°F.
GUIDED PRACTICE
11.
for Example 6
Mars has a maximum temperature of 7°C at the
equator and a minimum temperature of –133°C at
the winter pole.
• Write and solve a compound inequality that
describes the possible temperatures (in
degree Fahrenheit) on Mars.
ANSWER
–133 ≤
5
(F– 32) ≤ 27; 207.4 ≤ F ≤ 80.6
9
GUIDED PRACTICE
for Example 6
• Graph your solution. Then identify three
possible temperatures (in degrees
Fahrenheit) on Mars.
ANSWER
Sample answer: 100°F, 0°F, 25°F