Download 3. 2.

3-1 Graphing and Writing Inequalities
Warm Up
Compare. Write <, >, or =.
1. −3 < 2
2. 6.5 > 6.3
3.
4. 0.25 =
>
Tell whether the inequality x < 5 is true
or false for the following values of x.
5. x = –10
T
6. x = 5
7. x = 4.99
T
8. x =
Holt Algebra 1
F
T
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
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
Check It Out! Example 1
Graph each inequality.
a. c > 2.5
b. 22 – 4 ≥ w
c. m ≤ –3
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Example 2: Writing an Inequality from a Graph
Write the inequality shown by each graph.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Check It Out! Example 2
Write the inequality shown by the graph.
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 3: 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.
Holt Algebra 1
3-1 Graphing and Writing Inequalities
Check It Out! Example 3
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.
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.
Solving one-step inequalities is much like
solving one-step equations. To solve an
inequality, you need to isolate the variable using
the properties of inequality and inverse
operations.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Helpful Hint
Use an inverse operation to “undo” the
operation in an inequality. If the inequality
contains addition, use subtraction to undo
the addition.
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
Holt Algebra 1
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
Holt Algebra 1
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
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 1
Solve each inequality and graph the solutions.
a. s + 1 ≤ 10
b.
Holt Algebra 1
> –3 + t
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 1c
Solve the inequality and graph the solutions.
q – 3.5 < 7.5
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Since there can be an infinite number of solutions to
an inequality, it is not possible to check all the
solutions. You can check the endpoint and the
direction of the inequality symbol.
The solutions of x + 9 < 15 are given by x < 6.
Holt Algebra 1
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.
1
Understand the problem
The answer will be an inequality and a graph
that show all the possible amounts of money
that Sami can spend.
List important information:
• Sami can spend up to, or at most $30.
• Sami has already spent $14.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Example 2 Continued
2
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
3
Solve
g + 14 ≤ 30
– 14 – 14
g + 0 ≤ 16
Since 14 is added to g, subtract
14 from both sides to undo the
addition.
g ≤ 16
Draw a solid circle at 0 and16.
0
2
4
6
Holt Algebra 1
8 10 12 14 16 18 10
Shade all numbers greater than
0 and less than 16.
Solving Inequalities by
3-2 Adding or Subtracting
Example 2 Continued
4
Look Back
Check
Check the endpoint, 16.
g + 14 = 30
16 + 14 30
30 30 
Check a number less
than 16.
g + 14 ≤ 30
6 + 14 ≤ 30
20 ≤ 30
Sami can spend from $0 to $16.
Holt Algebra 1
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
1
Understand the problem
The answer will be an inequality and a graph
that show all the possible amounts of iron that
Sami can consume to reach the RDA.
List important information:
• The RDA of iron for Sarah is 15 mg.
• So far today she has consumed 11 mg.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 2 Continued
2
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
3
Solve
11 + x  15
–11
–11
x4
0
1
2
3
4
5
6
7 8
9 10
Since 11 is added to x,
subtract 11 from both
sides to undo the addition.
Draw a solid circle at 4.
Shade all numbers less
than 4.
x  4. Sarah can consume 4 mg or less of iron
without exceeding the RDA.
Holt Algebra 1
Solving Inequalities by
3-2 Adding or Subtracting
Check It Out! Example 2 Continued
4
Look Back
Check
Check the endpoint, 4.
Check a number less
than 4.
11 + x = 15
11 + 4 15
15 15 
11 + 3  15
11 + 3  15
14  15 
Sarah can consume 4 mg or less of iron
without exceeding the RDA.
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.
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
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Example 1B: Multiplying or Dividing by a Positive
Number
Solve the inequality and graph the solutions.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Check It Out! Example 1a
Solve the inequality and graph the solutions.
4k > 24
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Check It Out! Example 1b
Solve the inequality and graph the solutions.
–50 ≥ 5q
Holt Algebra 1
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
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
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Example 2B: Multiplying or Dividing by a Negative
Number
Solve the inequality and graph the solutions.
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Check It Out! Example 2
Solve each inequality and graph the solutions.
a. 10 ≥ –x
b. 4.25 > –0.25h
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
Holt Algebra 1
Solving Inequalities by
3-3 Multiplying or Dividing
Check It Out! Example 3
A pitcher holds 128 ounces of juice. What are
the possible numbers of 10-ounce servings that
one pitcher can fill?
Let x represent the number of servings of juice the
pitcher can contain.
10 oz
10
Holt Algebra 1
times
number of
servings
is at most
128 oz
•
x
≤
128
Solving Inequalities by
3-3 Multiplying or Dividing
Check It Out! Example 3 Continued
10x ≤ 128
Holt Algebra 1