Transcript Chapter 3 PPT- Algebrax

3-1 Graphing and Writing Inequalities
Warm Up
Solve.
1. 3x + 2 = 8
2.
3. -2k + 7 = -3
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Objectives
Identify solutions of inequalities with one
variable.
Write and graph inequalities with one variable.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
An inequality is a statement that two quantities
are not equal. The quantities are compared by
using the following signs:
≥
≠
A≤B
A≥B
A≠B
A is less
than or
equal to B.
A is greater
than or
equal to B.
A is not
equal to B.
<
>
≤
A<B
A>B
A is less
than B.
A is greater
than B.
A solution of an inequality is any value that
makes the inequality true.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Example 1: Identifying Solutions of Inequalities
Describe the solutions of x – 6 ≥ 4 in words.
–3
–9
x
x–6
?
?
x – 6 ≥ 4 –9 4
≥
Solution?
No
0
–6
?
9.9
3.9
?
10
4
?
–6 ≥4 3.9 ≥4 4 ≥4
Yes
No
No
10.1
4.1
?
12
6
?
4.1 ≥4 6 ≥4
Yes
Yes
When the value of x is a number less than 10, the value of x – 6 is
less than 4.
When the value of x is 10, the value of x – 6 is equal to 4.
When the value of x is a number greater than 10, the value of x – 6
is greater than 4.
It appears that the solutions of x – 6 ≥ 4 are all real numbers
greater than or equal to 10.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
An inequality like 3 + x < 9
has too many solutions to
list. You can use a graph on
a number line to show all
the solutions.
The solutions are shaded and an arrow shows that
the solutions continue past those shown on the
graph. To show that an endpoint is a solution, draw a
solid circle at the number. To show an endpoint is
not a solution, draw an empty circle.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Example 2: Graphing Inequalities
Graph each inequality.
A. m ≥
–
0
1
2
3
3
8
10 12
B. t < 5(–1 + 3)
t < 5(–1 + 3)
t < 5(2)
t < 10
–8 –6 –4 –2 0
Holt Algebra 1
2
4
6
3-1 Graphing and Writing Inequalities
Example 3: Writing an Inequality from a Graph
Write the inequality shown by each graph.
x<2
Use any variable. The arrow points to the left, so use
either < or ≤. The empty circle at 2 means that 2 is
not a solution, so use <.
x ≥ –0.5
Use any variable. The arrow points to the right, so
use either > or ≥. The solid circle at –0.5 means
that –0.5 is a solution, so use ≥.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Reading Math
“No more than” means “less than or
equal to.”
“At least” means “greater than or equal
to”.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Example 4: Application
Ray’s dad told him not to turn on the air
conditioner unless the temperature is at least
85°F. Define a variable and write an inequality
for the temperatures at which Ray can turn on
the air conditioner. Graph the solutions.
Let t represent the temperatures at which Ray can
turn on the air conditioner.
Turn on the AC when temperature
t
≥
t  85
70
75
Holt Algebra 1
80
85
is at least 85°F
90
85
Draw a solid circle at 85. Shade
all numbers greater than 85 and
draw an arrow pointing to the
right.
3-1 Graphing and Writing Inequalities
Check It Out! Example 4
A store’s employees earn at least $8.50 per
hour. Define a variable and write an
inequality for the amount the employees
may earn per hour. Graph the solutions.
Let w represent an employee’s wages.
An employee earns
at least
w
≥
w ≥ 8.5
−2 0
Holt Algebra 1
2 4
8.5
6
8 10 12 14 16 18
$8.50
8.50
Solving Inequalities by
3-2 Adding or Subtracting
Warm-Up
1. Describe the solutions of 7 < x + 4.
2. Graph h ≥ –4.75
–4.75
–5
–4.5
Write the inequality shown by each graph.
3.
4.
5. A cell phone plan offers free minutes for no more
than 250 minutes per month. Define a variable
and write an inequality for the possible number of
free minutes. Graph the solution.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Objectives
Solve one-step inequalities by using addition.
Solve one-step inequalities by using
subtraction.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Example 1A: Using Addition and Subtraction to Solve
Inequalities
Solve the inequality and graph the solutions.
x + 12 < 20
x + 12 < 20
–12 –12
x+0 < 8
x < 8
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Solving Inequalities by
3-2 Adding or Subtracting
Example 1B: Using Addition and Subtraction to Solve
Inequalities
Solve the inequality and graph the solutions.
d – 5 > –7
d – 5 > –7
+5 +5
d + 0 > –2
d > –2
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Solving Inequalities by
3-2 Adding or Subtracting
Example 1C: Using Addition and Subtraction to Solve
Inequalities
Solve the inequality and graph the solutions.
0.9 ≥ n – 0.3
0.9 ≥ n – 0.3
+0.3
+0.3
1.2 ≥ n – 0
1.2 ≥ n
1.2
0
Holt Algebra 1
1

2
Solving Inequalities by
3-2 Adding or Subtracting
Example 2: Problem-Solving Application
Sami has a gift card. She has already
used $14 of the of the total value, which
was $30. Write, solve, and graph an
inequality to show how much more she
can spend.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Example 2 Continued
Make a Plan
Write an inequality.
Let g represent the remaining amount of
money Sami can spend.
Amount
remaining
g
plus
amount
used
+
14
g + 14 ≤ 30
Holt Algebra 1
is at
most
≤
$30.
30
Solving Inequalities by
3-2 Adding or Subtracting
Example 2 Continued
Solve
g + 14 ≤ 30
– 14 – 14
g + 0 ≤ 16
g ≤ 16
0
2
4
6
Holt Algebra 1
8 10 12 14 16 18 10
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 2
The Recommended Daily Allowance (RDA)
of iron for a female in Sarah’s age group
(14-18 years) is 15 mg per day. Sarah has
consumed 11 mg of iron today. Write and
solve an inequality to show how many more
milligrams of iron Sarah can consume
without exceeding RDA.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 2 Continued
Make a Plan
Write an inequality.
Let x represent the amount of iron Sarah
needs to consume.
Amount
taken
11
plus
+
11 + x  15
Holt Algebra 1
amount
needed
x
is at
most
15 mg

15
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 2 Continued
Solve
11 + x  15
–11
–11
x4
0
1
2
3
4
5
6
7 8
9 10
x  4. Sarah can consume 4 mg or less of iron
without exceeding the RDA.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Warm-Up
Solve each inequality and graph the solutions.
1. 13 < x + 7
x>6
2. –6 + h ≥ 15
h ≥ 21
3. 6.7 + y ≤ –2.1
y ≤ –8.8
4. A certain restaurant has room for 120
customers. On one night, there are 72
customers dining. Write and solve an
inequality to show how many more
people can eat at the restaurant.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Objectives
Solve one-step inequalities by using
multiplication.
Solve one-step inequalities by using division.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Remember, solving inequalities is similar to
solving equations. To solve an inequality that
contains multiplication or division, undo the
operation by dividing or multiplying both sides of
the inequality by the same number.
The following rules show the properties of
inequality for multiplying or dividing by a
positive number. The rules for multiplying or
dividing by a negative number appear later in
this lesson.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Example 1A: Multiplying or Dividing by a Positive
Number
Solve the inequality and graph the solutions.
7x > –42
7x > –42
>
1x > –6
x > –6
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Solving Inequalities by
3-3 Multiplying or Dividing
Example 1B: Multiplying or Dividing by a Positive
Number
Solve the inequality and graph the solutions.
3(2.4) ≤ 3
7.2 ≤ m(or m ≥ 7.2)
0
2
4
Holt Algebra 1
6
8 10 12 14 16 18 20
Solving Inequalities by
3-3 Multiplying or Dividing
Example 1C: Multiplying or Dividing by a Positive
Number
Solve the inequality and graph the solutions.
r < 16
0
2
4
6
Holt Algebra 1
8 10 12 14 16 18 20
Solving Inequalities by
3-3 Multiplying or Dividing
If you multiply or divide both sides of an
inequality by a negative number, the resulting
inequality is not a true statement. You need to
reverse the inequality symbol to make the
statement true.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Caution!
Do not change the direction of the inequality
symbol just because you see a negative
sign. For example, you do not change the
symbol when solving 4x < –24.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Example 2A: Multiplying or Dividing by a Negative
Number
Solve the inequality and graph the solutions.
–12x > 84
x < –7
–7
–14 –12 –10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
Solving Inequalities by
3-3 Multiplying or Dividing
Example 2B: Multiplying or Dividing by a Negative
Number
Solve the inequality and graph the solutions.
24  x (or x  24)
10 12 14 16 18 20 22 24 26 28 30
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Example 3: Application
Jill has a $20 gift card to an art supply store
where 4 oz tubes of paint are $4.30 each after
tax. What are the possible numbers of tubes
that Jill can buy?
Let p represent the number of tubes of paint that Jill
can buy.
$4.30
times
4.30
•
Holt Algebra 1
number of tubes
is at most
$20.00.
p
≤
20.00
Solving Inequalities by
3-3 Multiplying or Dividing
Example 3 Continued
4.30p ≤ 20.00
Since p is multiplied by 4.30,
divide both sides by 4.30. The
symbol does not change.
p ≤ 4.65…
Since Jill can buy only whole numbers of tubes,
she can buy 0, 1, 2, 3, or 4 tubes of paint.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Warm-Up
Solve each inequality and graph the solutions.
1. 8x < –24 x < –3
2. –5x ≥ 30
x ≤ –6
3.
4.
x≥6
x > 20
5. A soccer coach plans to order more shirts for
her team. Each shirt costs $9.85. She has $77
left in her uniform budget. What are the
possible number of shirts she can buy?
0, 1, 2, 3, 4, 5, 6, or 7 shirts
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Warm-Up
Solve each inequality and graph the solutions.
1. 8x < –24
3.
x < –3
x > 20
2. –5x ≥ 30
x ≤ –6
4.
x≥6
5. A soccer coach plans to order more shirts for
her team. Each shirt costs $9.85. She has $77
left in her uniform budget. What are the
possible number of shirts she can buy?
0, 1, 2, 3, 4, 5, 6, or 7 shirts
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Objective
Solve inequalities that contain more than one
operation.
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Inequalities that contain more than one
operation require more than one step to solve.
Use inverse operations to undo the operations
in the inequality one at a time.
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Example 1A: Solving Multi-Step Inequalities
Solve the inequality and graph the solutions.
45 + 2b > 61
45 + 2b > 61
–45
–45
2b > 16
0
b>8
Holt Algebra 1
2
4
6
8 10 12 14 16 18 20
Solving Two-Step and
3-4 Multi-Step Inequalities
Example 1B: Solving Multi-Step Inequalities
Solve the inequality and graph the solutions.
8 – 3y ≥ 29
8 – 3y ≥ 29
–8
–8
–3y ≥ 21
–7
–10 –8 –6 –4 –2
y ≤ –7
Holt Algebra 1
0
2
4
6
8 10
Solving Two-Step and
3-4 Multi-Step Inequalities
Check It Out! Example 1c
Solve the inequality and graph the solutions.
1 – 2n ≥ 21
–1
–1
–2n ≥ 20
n ≤ –10
Holt Algebra 1
–10
–20
–16
–12
–8
–4
0
Solving Two-Step and
3-4 Multi-Step Inequalities
To solve more complicated inequalities, you
may first need to simplify the expressions on
one or both sides by using the order of
operations, combining like terms, or using the
Distributive Property.
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Example 2A: Simplifying Before Solving Inequalities
Solve the inequality and graph the solutions.
2 – (–10) > –4t
12 > –4t
–3 < t (or t > –3)
–3
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Solving Two-Step and
3-4 Multi-Step Inequalities
Example 2B: Simplifying Before Solving Inequalities
Solve the inequality and graph the solutions.
–4(2 – x) ≤ 8
−4(2 – x) ≤ 8
−4(2) − 4(−x) ≤
–8 + 4x ≤ 8
+8
+8
4x ≤ 16
–10 –8 –6 –4 –2
x≤4
Holt Algebra 1
0
2
4
6
8 10
Solving Two-Step and
3-4 Multi-Step Inequalities
Check It Out! Example 2b
Solve the inequality and graph the solutions.
3 + 2(x + 4) > 3
3 + 2(x + 4) > 3
3 + 2x + 8 > 3
2x + 11 > 3
– 11 – 11
2x
> –8
x > –4
Holt Algebra 1
–10 –8 –6 –4 –2
0
2
4
6
8 10
Solving Two-Step and
3-4 Multi-Step Inequalities
Check It Out! Example 2c
Solve the inequality and graph the solutions.
5 < 3x – 2
+2
+2
7 < 3x
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Check It Out! Example 2c Continued
Solve the inequality and graph the solutions.
7 < 3x
0
2
Holt Algebra 1
4
6
8
10
Solving Two-Step and
3-4 Multi-Step Inequalities
Example 3: Application
To rent a certain vehicle, Rent-A-Ride charges $55.00
per day with unlimited miles. The cost of renting a
similar vehicle at We Got Wheels is $38.00 per day plus
$0.20 per mile. For what number of miles is the cost at
Rent-A-Ride less than the cost at We Got Wheels?
Let m represent the number of miles. The cost for
Rent-A-Ride should be less than that of We Got
Wheels.
Cost at
Rent-ARide
must be
less
than
55
<
Holt Algebra 1
daily
cost at
We Got
Wheels
38
plus
+
$0.20
per mile
0.20
times
# of
miles.

m
Solving Two-Step and
3-4 Multi-Step Inequalities
Example 3 Continued
55 < 38 + 0.20m
Since 38 is added to 0.20m, subtract 8
55 < 38 + 0.20m
from both sides to undo the addition.
–38 –38
17 < 0.20m
Since m is multiplied by 0.20, divide
both sides by 0.20 to undo the
multiplication.
85 < m
Rent-A-Ride costs less when the number of miles is
more than 85.
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Check It Out! Example 3
The average of Jim’s two test scores must
be at least 90 to make an A in the class.
Jim got a 95 on his first test. What grades
can Jim get on his second test to make an
A in the class?
Let x represent the test score needed. The
average score is the sum of each score divided
by 2.
First
test
score
plus
(95
Holt Algebra 1
+
second
test
score
x)
divided
by

number
of scores
2
is greater
than or
equal to
≥
total
score
90
Solving Two-Step and
3-4 Multi-Step Inequalities
Check It Out! Example 3 Continued
Since 95 + x is divided by 2, multiply
both sides by 2 to undo the division.
95 + x ≥ 180
–95
–95
Since 95 is added to x, subtract 95 from
both sides to undo the addition.
x ≥ 85
The score on the second test must be 85 or higher.
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. 13 – 2x ≥ 21 x ≤ –4
2. –11 + 2 < 3p
p > –3
3. 23 < –2(3 – t)
t>7
4.
Holt Algebra 1
Solving Two-Step and
3-4 Multi-Step Inequalities
Lesson Quiz: Part II
5. A video store has two movie rental plans. Plan
A includes a $25 membership fee plus $1.25 for
each movie rental. Plan B costs $40 for
unlimited movie rentals. For what number of
movie rentals is plan B less than plan A?
more than 12 movies
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Warm Up
Solve each equation.
1. 2x = 7x + 15 x = –3
2. 3y – 21 = 4 – 2y
y=5
3. 2(3z + 1) = –2(z + 3) z = –1
4. 3(p – 1) = 3p + 2
no solution
5. Solve and graph 5(2 – b) > 52. b < –3
–6
Holt Algebra 1
–5
–4
–3
–2
–1
0
Inequalities with
3-5 Solving
Variables on Both Sides
Objective
Solve inequalities that contain variable
terms on both sides.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Example 1A: Solving Inequalities with Variables on
Both Sides
Solve the inequality and graph the solutions.
y ≤ 4y + 18
y ≤ 4y + 18
–y –y
0 ≤ 3y + 18
–18
– 18
–18 ≤ 3y
–10 –8 –6 –4 –2
–6 ≤ y (or y  –6)
Holt Algebra 1
0
2
4
6
8 10
Inequalities with
3-5 Solving
Variables on Both Sides
Example 1B: Solving Inequalities with Variables on
Both Sides
Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
–2m
– 2m
2m – 3 <
+6
+3
+3
2m
<
9
4
Holt Algebra 1
5
6
Inequalities with
3-5 Solving
Variables on Both Sides
Example 2: Business Application
The Home Cleaning Company charges $312 to
power-wash the siding of a house plus $12 for
each window. Power Clean charges $36 per
window, and the price includes power-washing
the siding. How many windows must a house
have to make the total cost from The Home
Cleaning Company less expensive than Power
Clean?
Let w be the number of windows.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Example 2 Continued
Home
Cleaning
Company
siding
charge
312
plus
+
$12 per
window
12
times
# of
windows
is
less
than
Power
Clean
cost per
window
•
w
<
36
312 + 12w < 36w
– 12w –12w
312 < 24w
times
# of
windows.
•
w
To collect the variable terms,
subtract 12w from both sides.
Since w is multiplied by 24, divide
both sides by 24 to undo the
multiplication.
13 < w
The Home Cleaning Company is less expensive for
houses with more than 13 windows.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Check It Out! Example 2
A-Plus Advertising charges a fee of $24 plus
$0.10 per flyer to print and deliver flyers. Print
and More charges $0.25 per flyer. For how
many flyers is the cost at A-Plus Advertising
less than the cost of Print and More?
Let f represent the number of flyers printed.
A-Plus
Advertising plus
fee of $24
24
+
Holt Algebra 1
$0.10
per
flyer
0.10
times
# of
flyers
is less
than
•
f
<
Print and
More’s cost
per flyer
0.25
times
•
# of
flyers.
f
Inequalities with
3-5 Solving
Variables on Both Sides
Check It Out! Example 2 Continued
24 + 0.10f < 0.25f
–0.10f –0.10f
24
To collect the variable terms,
subtract 0.10f from both sides.
< 0.15f
Since f is multiplied by 0.15,
divide both sides by 0.15 to
undo the multiplication.
160 < f
More than 160 flyers must be delivered to make
A-Plus Advertising the lower cost company.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
You may need to simplify one or both sides of
an inequality before solving it. Look for like
terms to combine and places to use the
Distributive Property.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Check It Out! Example 3a
Solve the inequality and graph the solutions.
5(2 – r) ≥ 3(r – 2)
16 ≥ 8r
5(2 – r) ≥ 3(r – 2)
5(2) – 5(r) ≥ 3(r) + 3(–2)
10 – 5r ≥ 3r – 6
+6
+6
16 − 5r ≥ 3r
+ 5r +5r
16
Holt Algebra 1
≥ 8r
2≥r
–6
–4
–2
0
2
4
Inequalities with
3-5 Solving
Variables on Both Sides
There are special cases of inequalities called
identities and contradictions.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Example 4A: Identities and Contradictions
Solve the inequality.
2x – 7 ≤ 5 + 2x
2x – 7 ≤ 5 + 2x
–2x
–2x
–7 ≤ 5
The inequality 2x − 7 ≤ 5 + 2x is an identity. All
values of x make the inequality true. Therefore,
all real numbers are solutions.
Holt Algebra 1
Inequalities with
3-5 Solving
Variables on Both Sides
Example 4B: Identities and Contradictions
Solve the inequality.
2(3y – 2) – 4 ≥ 3(2y + 7)
2(3y – 2) – 4 ≥ 3(2y + 7)
2(3y) – 2(2) – 4 ≥ 3(2y) + 3(7)
6y – 4 – 4 ≥ 6y + 21
6y – 8 ≥ 6y + 21
–6y
–6y
–8 ≥ 21
No values of y make the inequality true.
There are no solutions.
Holt Algebra 1
3-6 Solving Compound Inequalities
Warm-Up
Solve each inequality and graph the solutions.
1. t < 5t + 24
2. 5x – 9 ≤ 4.1x – 81
3. 4b + 4(1 – b) > b – 9
Solve each inequality.
4. 2y – 2 ≥ 2(y + 7)
5. 2(–6r – 5) < –3(4r + 2)
Holt Algebra 1
3-6 Solving Compound Inequalities
Objectives
Solve compound inequalities with one
variable.
Graph solution sets of compound inequalities
with one variable.
Holt Algebra 1
3-6 Solving Compound Inequalities
The inequalities you have seen so far are
simple inequalities. When two simple
inequalities are combined into one statement
by the words AND or OR, the result is called a
compound inequality.
Holt Algebra 1
3-6 Solving Compound Inequalities
Holt Algebra 1
3-6 Solving Compound Inequalities
Example 1: Chemistry Application
The pH level of a popular shampoo is between 6.0
and 6.5 inclusive. Write a compound inequality to
show the pH levels of this shampoo. Graph the
solutions.
Let p be the pH level of the shampoo.
6.0
is less than
or equal to
pH level
is less than
or equal to
6.5
6.0
≤
p
≤
6.5
6.0 ≤ p ≤ 6.5
5.9
6.0
Holt Algebra 1
6.1
6.2
6.3
6.4
6.5
3-6 Solving Compound Inequalities
In this diagram, oval A represents some integer
solutions of x < 10 and oval B represents some
integer solutions of x > 0. The overlapping region
represents numbers that belong in both ovals. Those
numbers are solutions of both x < 10 and x > 0.
Holt Algebra 1
3-6 Solving Compound Inequalities
You can graph the solutions of a compound
inequality involving AND by using the idea of an
overlapping region. The overlapping region is
called the intersection and shows the numbers
that are solutions of both inequalities.
Holt Algebra 1
3-6 Solving Compound Inequalities
Example 2A: Solving Compound Inequalities Involving
AND
Solve the compound inequality and graph
the solutions.
–5 < x + 1 < 2
–5 < x + 1 < 2
–1
–1–1
–6 < x < 1
Graph –6 < x.
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Graph x < 1.
Graph the intersection by
finding where the two
graphs overlap.
3-6 Solving Compound Inequalities
Example 2B: Solving Compound Inequalities Involving
AND
Solve the compound inequality and graph
the solutions.
8 < 3x – 1 ≤ 11
8 < 3x – 1 ≤ 11
+1
+1 +1
9 < 3x ≤ 12
3<x≤4
Holt Algebra 1
3-6 Solving Compound Inequalities
Example 2B Continued
Graph 3 < x.
Graph x ≤ 4.
–5 –4 –3 –2 –1
Holt Algebra 1
0
1
2
3
4
5
Graph the intersection by
finding where the two
graphs overlap.
3-6 Solving Compound Inequalities
Check It Out! Example 2a
Solve the compound inequality and graph the
solutions.
–9 < x – 10 < –5
–9 < x – 10 < –5
+10
+10 +10
1<x<5
Graph 1 < x.
Graph x < 5.
–5 –4 –3 –2 –1
Holt Algebra 1
0
1
2
3
4
5
Graph the intersection by
finding where the two
graphs overlap.
3-6 Solving Compound Inequalities
In this diagram, circle A represents some integer
solutions of x < 0, and circle B represents some
integer solutions of x > 10. The combined shaded
regions represent numbers that are solutions of
either x < 0 or x >10.
Holt Algebra 1
3-6 Solving Compound Inequalities
You can graph the solutions of a compound
inequality involving OR by using the idea of
combining regions. The combine regions are called
the union and show the numbers that are
solutions of either inequality.
>
Holt Algebra 1
3-6 Solving Compound Inequalities
Example 3A: Solving Compound Inequalities Involving
OR
Solve the inequality and graph the solutions.
8 + t ≥ 7 OR 8 + t < 2
8 + t ≥ 7 OR 8 + t < 2
–8
–8
–8
−8
t ≥ –1 OR
t < –6
Graph t ≥ –1.
Graph t < –6.
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Graph the union by
combining the regions.
3-6 Solving Compound Inequalities
Example 3B: Solving Compound Inequalities Involving
OR
Solve the inequality and graph the solutions.
4x ≤ 20 OR 3x > 21
4x ≤ 20 OR 3x > 21
x ≤ 5 OR x > 7
Graph x ≤ 5.
Graph x > 7.
–10 –8 –6 –4 –2
Holt Algebra 1
0
2
4
6
8 10
Graph the union by
combining the regions.
3-6 Solving Compound Inequalities
Check It Out! Example 3a
Solve the compound inequality and graph the
solutions.
2 +r < 12 OR r + 5 > 19
2 +r < 12 OR r + 5 > 19
–2
–2
–5 –5
Solve each simple
inequality.
r < 10 OR r > 14
Graph r < 10.
Graph r > 14.
–4 –2 0
Holt Algebra 1
2
4
6
8 10 12 14 16
Graph the union by
combining the regions.
3-6 Solving Compound Inequalities
Example 4A: Writing a Compound Inequality
Graph
from a
Write the compound inequality shown by the graph.
The shaded portion of the graph is not between two values, so
the compound inequality involves OR.
On the left, the graph shows an arrow pointing left, so use
either < or ≤. The solid circle at –8 means –8 is a solution so
use ≤. x ≤ –8
On the right, the graph shows an arrow pointing right, so use
either > or ≥. The empty circle at 0 means that 0 is not a
solution, so use >. x > 0
The compound inequality is x ≤ –8 OR x > 0.
Holt Algebra 1
3-6 Solving Compound Inequalities
Example 4B: Writing a Compound Inequality from a
Graph
Write the compound inequality shown by the graph.
The shaded portion of the graph is between the values –2 and
5, so the compound inequality involves AND.
The shaded values are on the right of –2, so use > or ≥. The
empty circle at –2 means –2 is not a solution, so use >.
m > –2
The shaded values are to the left of 5, so use < or ≤. The
empty circle at 5 means that 5 is not a solution so use <.
m<5
The compound inequality is m > –2 AND m < 5
(or -2 < m < 5).
Holt Algebra 1
3-6 Solving Compound Inequalities
Lesson Quiz: Part I
1. The target heart rate during exercise for a 15
year-old is between 154 and 174 beats per
minute inclusive. Write a compound inequality to
show the heart rates that are within the target
range. Graph the solutions.
154 ≤ h ≤ 174
Holt Algebra 1
3-6 Solving Compound Inequalities
Lesson Quiz: Part II
Solve each compound inequality and graph
the solutions.
2. 2 ≤ 2w + 4 ≤ 12
–1 ≤ w ≤ 4
3. 3 + r > −2 OR 3 + r < −7
r > –5 OR r < –10
Holt Algebra 1
3-6 Solving Compound Inequalities
Lesson Quiz: Part III
Write the compound inequality shown by
each graph.
4.
x < −7 OR x ≥ 0
5.
−2 ≤ a < 4
Holt Algebra 1
3-6 Solving Compound Inequalities
Warm-Up
1. The target heart rate during exercise for a 15
year-old is between 154 and 174 beats per
minute inclusive. Write a compound inequality to
show the heart rates that are within the target
range. Graph the solutions.
Solve each compound inequality and graph
the solutions.
2. 2 ≤ 2w + 4 ≤ 12
3. 3 + r > −2 OR 3 + r < −7
Holt Algebra 1
3-6 Solving Compound Inequalities
Objectives
Solve inequalities in one variable involving
absolute value expressions.
Holt Algebra 1
3-6 Solving Compound Inequalities
When an inequality contains an absolute-value
expression, it can be rewritten as a compound
inequality. The inequality |x| < 5 describes all
real number whose distance from 0 is less
than 5 units. The solutions can be written as
-5 < x < 5.
Holt Algebra 1
3-6 Solving Compound Inequalities
Example: Solving Absolute Value Inequalities
involving <.
Solve the inequality and graph the solutions.
|x| + 3 < 12
–3
–3
|x| < 9
Holt Algebra 1
|x + 4| ≤ 2
x+4≤2
-4 -4
x + 4 ≥ -2
-4 -4
x ≤ -2 and
x ≥ -6
3-6 Solving Compound Inequalities
The inequality |x| > 5 describes all real
number whose distance from 0 is greater than
5 units. The solutions are all numbers less
than -5 or greater than 5. The solutions can
be written as x < -5 OR x > 5.
Holt Algebra 1
3-6 Solving Compound Inequalities
Example: Solving Absolute Value Inequalities
involving >.
Solve the inequality and graph the solutions.
|x| - 20 > -13
+ 20
+20
|x| > 7
x < -7 or x > 7
|x - 8|+5 ≥ 11
- 5 -5
|x – 8| ≥ 6
x- 8≥6
+8 +8
x - 8 ≤ -6
+8 +8
x ≥ 14 or
Holt Algebra 1
x≤2