6.4 Solve Compound Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
6.4
Warm-Up
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
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
6.4
Example 1
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:
6.4
Guided Practice
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
6.4
Example 2
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.
6.4
Example 2
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.
6.4
Guided Practice
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
6.4
Example 3
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.
6.4
Example 3
ANSWER
The solutions are all real numbers greater than –3 and
less than 4.
Graph:
6.4
Guided Practice
Solve the inequality. Graph your solution.
4. –7 < x – 5 < 4
ANSWER
–2 < x < 9
9
Graph:
–6
5.
–4
–2
0
2
4
6
10 ≤ 2y + 4 ≤ 24
3 ≤ y ≤ 10
ANSWER
3
Graph:
0
6.
2
4
–7 < –z – 1 < 3
ANSWER
–4 < z < 6
6
8
10
12
8
10
6.4
Example 4
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
–5 ≤ x ≤ 2
Simplify.
Multiply each
expression by –1 and
reverse both inequality
symbols.
Simplify.
Rewrite in the form
a ≤ x ≤ b.
6.4
Example 4
ANSWER
The solutions are all real
numbers greater than or
equal to –5 and less than
or equal to 2.
6.4
Guided Practice
Solve the inequality. Graph your solution.
7.
–14 < x – 8 < –1
ANSWER
8.
–6 < x < 7
–1 ≤ –5t + 2 ≤ 4
ANSWER
– 2 ≤t≤ 3
5
5
6.4
Example 5
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.
6.4
Example 5
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.
6.4
Guided Practice
Solve the inequality. Graph your solution.
9.
3h + 1< – 5 or
ANSWER
10.
h < –2 or h > 6
4c + 1 ≤ –3 or
ANSWER
2h – 5 > 7
5c – 3 > 17
c ≤ –1 or c > 4
6.4
Example 6
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.
6.4
Example 6
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
6.4
Example 6
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
6.4
Example 6
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.
6.4
Guided Practice
11.
Mars has a maximum temperature of 27°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.
–133 ≤ 5 (F– 32) ≤ 27; -207.4 ≤ F ≤ 80.6
ANSWER
9
• Graph your solution. Then identify three
possible temperatures (in degrees
Fahrenheit) on Mars.
ANSWER
Sample answer:
-100°F, 0°F, 25°F
6.4
1.
Lesson Quiz
Solve x + 4 < 7 or 2x – 5 > 3. Graph your solution.
ANSWER
all real numbers less than 3 or greater
than 4
2. Solve 3x + 2 > –7 and 4x – 1 < –5. Graph your solution.
ANSWER
all real numbers greater than –3 and
less than –1
6.4
3.
Lesson Quiz
The smallest praying mantis is 0.4 inch in length.
The largest is 6 inches. Write and graph a
compound inequality that describes the possible
lengths L of a praying mantis.
ANSWER
0.4 <
– L <
–6