Download Solving Inequalities by Multiplying or Dividing

2-3
Solving Inequalities by
Multiplying or Dividing
Going Deeper
Essential question: How can you use properties to justify solutions to inequalities
that involve multiplication and division?
TEACH
Standards for
Mathematical Content
1
A-REI.2.3 Solve linear…inequalities…in one
variable…
eXplore
Questioning Strategies
• Why do you need to reverse the inequality symbol
when multiplying or dividing by a negative
number? Multiplying or dividing by a negative
Prerequisites
Solving Equations by Multiplying or Dividing
Graphing and Writing Inequalities
number without reversing the inequality symbol
results in a false inequality. You must reverse the
symbol to get a true inequality.
Math Background
Inequalities involving multiplication and division
can be solved using inverse operations to isolate
the variable in a manner similar to equations
involving multiplication and division. The solution
steps are justified by using the Multiplication and
Division Properties of Inequality. However, one
difference between solving such inequalities and
equations exists. When both sides of an inequality
are multiplied or divided by a negative number,
the inequality sign must be reversed. As with
solutions of inequalities that involved addition and
subtraction, solutions of inequalities that involve
multiplication and division can be written using set
notation and graphed on a number line.
Differentiated Instruction
To help visual learners understand the effect of
multiplying or dividing both sides of an inequality
by a negative number, demonstrate the effect of
multiplying or dividing by -1. Sketch a number
line with points at -3 and 2. Ask students what
inequality compares the numbers. With a different
color, plot the points that represent the product of
(-1)(-3) and (-1)(2). Ask students what inequality
compares the numbers. Emphasize how they
needed to reverse the inequality sign. As you carry
out these steps, record the work as follows:
INTR O D U C E
(-1)(-3) ? (-1)(2)
Review solving a multiplication equation and a
division equation, such as 6x = -42 and ​ __x ​  = 4,
3
reminding students of the properties of equality
that justify the solution steps. Tell students that
they can solve inequalities involving multiplication
and division in a similar manner, with one
difference that they will discover as they work
through the Explore section. Point out that they
will represent solutions of inequalities involving
multiplication and division in the same ways as
they did solutions of inequalities that involve
addition and subtraction: using set notation and
graphing on a number line.
3 > -2
2
or
2 > -3
(-1)(2) ? (-1)(-3)
-2 < 3
EXAMPLE
Questioning Strategies
• Why is the Multiplication Property of Inequality
used to solve the inequalities? The inequalities
involve division. Because multiplication is the
inverse of division, multiplication can be used to
isolate the variable.
• Why does the inequality symbol stay the same in
part A but change in part B? When you multiply
both sides of an inequality by a positive number,
as in part A, the inequality symbol stays the
same. When you multiply both sides of an
inequality by a negative number, as in part B, you
must reverse the inequality symbol.
continued
Chapter 2 85
Lesson 3
© Houghton Mifflin Harcourt Publishing Company
-3 < 2
Name
Class
Date
Notes
2-3
Solving Inequalities by Multiplying
or Dividing
Going Deeper
Essential question: How can you use properties to justify solutions to inequalities that
involve multiplication and division?
A-REI.2.3
1
EXPLORE
Multiplying or Dividing by a Negative Number
The following two inequalities are true.
4<5
15 > 12
What happens to the inequalities if you multiply both sides of the first inequality by 4 and
divide both sides of the second inequality by 3?
4<5
15 > 12
4(4) < 4(5)
15
12
___
> ___
3
3
16 < 20
5> 4
Both statements are still true: 16 is less than 20, and 5 is greater than 4.
© Houghton Mifflin Harcourt Publishing Company
Now, multiply the first inequality by -4 and divide the second inequality by -3. Do not
change the inequality symbol when you do these multiplications.
4<5
15 > 12
-4(4) < -4(5)
15
12
___
> ___
-3
-3
-16 < -20
-5 > -4
Is -16 less than -20? No, -16 is closer to 0 than -20 is, so it is greater than -20.
Is -5 greater than -4? No, -5 is farther from 0 than -4, so it is less than -4.
Repeat the multiplication by -4 and the division by -3, but this time reverse the
inequality symbol when you do.
4<5
15 > 12
-4(4) > -4(5)
15
12
___
< ___
-3
-3
-5 < -4
-16 > -20
Yes
Do you get a true statement in each case?
REFLECT
1a. When solving inequalities, if you multiply by a negative number, you must
reverse the inequality symbol
1b. When solving inequalities, if you divide by a negative number, you must
reverse the inequality symbol
Chapter 2
85
Lesson 3
You can use the following inequality properties to solve inequalities involving
multiplication and division. These properties are also true for ≥ and ≤.
If a > b and c > 0, then ac > bc.
If a < b and c > 0, then ac < bc.
If a > b and c < 0, then ac < bc.
If a < b and c < 0, then ac > bc.
Multiplication Property of Inequality
a
b
__
If a > b and c > 0, then __
c > c.
a
b
__
If a < b and c > 0, then __
c < c.
a
b
__
If a > b and c < 0, then __
c < c.
a
b
__
If a < b and c < 0, then __
c > c.
A-REI.2.3
2
EXAMPLE
Multiplying to Find the Solution Set
Solve. Write the solution using set notation. Graph your solution.
x>3
__
A
2
2
x >
__
(2)
x>
Solution set:
Multiplication Property of Inequality
2 (3)
6
Simplify.
{x x > 6}
-12 -10 -8 -6 -4 -2
0
2
4
6
8
10
12 14
x ≤ -2
___
B
-4
-4
x ≥
( ___
-4 )
Multiplication Property of Inequality;
-4 (-2)
reverse
x≥
Solution set:
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© Houghton Mifflin Harcourt Publishing Company
Division Property of Inequality
8
≤ symbol.
Simplify.
{x x ≥ 8}
-24 -20 -16 -12 -8 -4
0
4
8
12
16
20
24 28
REFLECT
2a. Suppose the inequality symbol in Part A had been ≥. Describe the solution set.
The solution set would have been 6 and all values greater than 6.
2b. Suppose the inequality symbol in Part B had been <. Describe the solution set.
The solution set would have been all values greater than 8.
Chapter 2
Chapter 2
86
Lesson 3
86
Lesson 3
2 EXAMPLE
Avoid Common Errors
Students may reverse the inequality symbol
when multiplying or dividing both sides of an
inequality by any number. They may also reverse
the inequality symbol when they add to or subtract
from both sides of an inequality. Encourage
students to check their solutions by substituting
a value from the solution set into the original
inequality.
continued
EXTRA EXAMPLE
Solve. Write the solution using set notation. Graph
your solution.
x < -2 {x | x < -8}; the graph has an empty
A. __
4
circle on -8, and the line to the left of -8 is
shaded.
x ≥ 3 {x | x ≤ -9}; the graph has a solid circle
B. ___
-3
on -9, and the line to the left of -9 is shaded.
3
CLOS E
Essential Question
How can you use properties to justify solutions
to inequalities that involve multiplication and
division?
EXAMPLE
Questioning Strategies
• How do the inequalities in this example differ
from those in 2 EXAMPLE ? These inequalities
Use the Multiplication Property of Inequality
to justify multiplying both sides of a division
inequality by the same number to isolate the
variable. Use the Division Property of Inequality
to justify dividing both sides of a multiplication
inequality by the same number to isolate the
variable. When multiplying or dividing by a
negative number, reverse the inequality symbol.
involve multiplication instead of division.
• How is solving these inequalities similar to
solving those in 2 EXAMPLE ? The inverse
of the operation involved in the inequalities is
used to isolate the variable and the inequality
symbol is reversed when a negative number is
used in the inverse operation.
• How does solving these inequalities differ from
solving those in 2 EXAMPLE ? The inverse
operation used is division instead of multiplication.
EXTRA EXAMPLE
Solve. Write the solution using set notation. Graph
your solution.
A. 7x ≤ -21 {x | x ≤ -3}; the graph has a solid
circle on -3, and the line to the left of -3 is
shaded.
B. -4x > 16 {x | x < -4}; the graph has an empty
circle on -4, and the line to the left of -4 is
shaded.
PR ACTICE
Where skills are
taught
Highlighting
the Standards
This lesson provides numerous opportunities
to address Mathematical Practices Standard
2 (Reason abstractly and quantitatively)
as students solve inequalities. As students
determine whether or not to reverse the
inequality symbol when multiplying
or dividing both sides of an inequality,
encourage them to think about the effect
of multiplying or dividing by a positive or
negative number and not just following a rule.
Chapter 2
87
Where skills are
practiced
2 EXAMPLE
EXS. 2, 3, 5
3 EXAMPLE
EXS. 1, 4
Lesson 3
© Houghton Mifflin Harcourt Publishing Company
Summarize
Have students write a journal entry in which they
describe how to solve an inequality involving
multiplication and an inequality involving division.
They should include the properties that justify the
steps in their description. Encourage students to
use a variety of inequality symbols and include
negative and positive numbers. Have students
write their solutions in set notation and represent
them with graphs on a number line. Have them
explain the decisions they had to make to graph the
solutions.
Notes
A-REI.2.3
3
EXAMPLE
Dividing to Find the Solution Set
Solve. Write the solution using set notation. Graph your solution.
3x ≥ -9
A
-9
3x ≥ ______
______
3
Division
Property of Inequality
3
x ≥ -3
Simplify.
Solution set: {x x ≥ -3}
-4 -3 -2 -1
0
1
2
3
4
5
6
7
8
9
-5x < 20
B
20
-5x > ______
______
-5
Division
Property of Inequality;
reverse
< symbol.
-5
x > -4
Solution set:
Simplify.
{x x > -4}
-4 -3 -2 -1
0
1
2
3
4
5
6
7
8
9
REFLECT
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3a. There is a negative number in both Parts A and B. Why is the inequality symbol
only reversed in Part B?
When solving an inequality, the inequality symbol is reversed only when both
sides are multiplied or divided by a negative number. This happens only when the
coefficient of the variable is a negative number.
3b. Suppose the inequality symbol in Part A had been >. Describe the solution set.
The solution set would have been all values greater than -3.
3c. Suppose the inequality symbol in Part B had been ≤. Describe the solution set.
The solution set would have been -4 and all values greater than -4.
Chapter 2
87
Lesson 3
PRACTICE
Solve. Justify your steps. Write each solution using set notation. Graph your
solution.
32
4x < __
__
4
4
x <8
-24 -20 -16 -12 -8 -4
0
4
8
12
16
20
24 28
15
20
25
30 35
16
20
24 28
Division Property of Inequality
Simplify.
{x x < 8}
x > -3
2. __
5
-30 -25 -20 -15 -10 -5
x > 5(-3)
5 __
5
( )
x > -15
0
5
10
Multiplication Property of Inequality
Simplify.
{x x > -15}
x ≤ -4
3. ___
-4
-24 -20 -16 -12 -8 -4
x ≥ -4(-4)
-4 ___
-4
( )
x ≥ 16
0
4
8
12
Mult. Property of Inequality; reverse inequality symbol.
Simplify.
{x x ≥ 16}
4. -2x ≥ -6
–4 –3 –2 –1
-6
-2x ≤ ___
____
-2
-2
x ≤3
0
1
2
3
4
5
6
7
8
9
Division Property of Inequality; reverse inequality symbol.
Simplify.
{x x ≤ 3}
x <3
5. ___
-6
x > -6(3)
-6 ___
-6
( )
x > -18
-24 -21 -18 -15 -12 -9
–6
–3
0
3
6
9
12 15
Mult. Property of Inequality; reverse inequality symbol.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
1. 4x < 32
Simplify.
{x x > -18}
Chapter 2
Chapter 2
88
Lesson 3
88
Lesson 3
ADD I T I O NA L P R AC TI C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. a > 8
2. y >-3
3. n ≤-12
4. c ≤ -24
5. y > 20
6. s ≤ -1.5
7. b > 18
© Houghton Mifflin Harcourt Publishing Company
8. z ≤ 2
9. 5p ≤ 16; p ≤ 3.2; 0, 1, 2, or 3 pieces
10. 5s ≤ 128; s ≤ 25.6; 0 to 25 cups
11. 4b ≥ 50; b ≥ 12.50; $12.50 each
Problem Solving
1. 0.50g ≤ 3; g ≤ 6; 0, 1, 2, 3, 4, 5, or 6
2. 15d ≤ 21; d ≤ 1.40; up to $1.40
3. 2.5h ≤ 7; h ≤ 2.8; up to 2.8 hours
4. 11q ≤ 50; q ≤ 4.54; 0, 1, 2, 3, or 4
5. A
6. G
7. A
Chapter 2
89
Lesson 3
Name
Class
Notes
2-3
Date
Additional Practice
© Houghton Mifflin Harcourt Publishing Company
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