Transcript Graphing and Writing Inequalities

Writing and Solving Inequalities
How can you represent relationships
using inequalities?
Vocabulary
An inequality is a statement that compares two
expressions that are not strictly equal by using
one of the following inequality signs.
Symbol
Meaning
<
is less than
≤
is less than or equal to
>
is greater than
≥
is greater than or equal to
≠
is not equal to
Vocabulary
A solution of an inequality is any value of the
variable that makes the inequality true. One way
you can find solutions is by using a table.
Writing and Solving Inequalities
Example 1
Mr. Bowker, being a teacher, can afford to spend
at most $50 for a birthday dinner at a
restaurant, including a 15% tip. Describe some
costs that are within his budget.
A. Which inequality symbol can be used to
represent “at most”? _________
B. Complete the verbal model for the situation.
Cost before tip
(dollars)
15%
Cost before tip
(dollars)
Budget limit
(dollars)
Writing and Solving Inequalities
Example 1
Mr. Bowker, being a teacher, can afford to spend
at most $50 for a birthday dinner at a
restaurant, including a 15% tip. Describe some
costs that are within his budget.
C. Write and simplify an inequality for the
model. _________________________
Writing and Solving Inequalities
Example 1
D. Complete the table to find some costs that
are within Mr. Bowker’s budget.
Cost
Substitute
Compare
Solution?
$47
1.15 47 ≤ 50
? $54.05 ≤ 50 ?
No
$45
1.15 45 ≤ 50
Writing and Solving Inequalities
Example 1
1. Can Mr. Bowker spend $40 on the meal
before the tip? Explain.
_____________________________________
_____________________________________
2. What whole dollar amount is the most Mr.
Bowker can spend before the tip? Explain.
_____________________________________
_____________________________________
Writing and Solving Inequalities
Example 1
3. The solution set of an equation or inequality
consists of all values that make the statement
true. Describe the whole dollar amounts that
are in the solution set for this situation.
_____________________________________
_____________________________________
Writing and Solving Inequalities
Example 1
4. Suppose Mr. Bowker also has to pay a 6%
meal tax. Write an inequality to represent the
new situation. Then identify two solutions.
_____________________________________
_____________________________________
_____________________________________
Summary
In your own words, describe what an inequality
is and what it means to have a solution to an
inequality.
What do inequalities have in common with
equations? How are they different?
Next lesson, we will learn how to solve
inequalities in a systematic way.
Solving Inequalities by Adding or
Subtracting
How can you use properties to justify
solutions to inequalities that involve
addition and subtraction?
Introduction
Suppose we had to solve the equation
𝑥 + 3 = −5
Recall that the Subtraction Property of Equality
justifies subtracting 3 from both sides which
gives us the solution 𝑥 = −8.
The solution can be represented as a point on
the number line.
Introduction
What inequality does the graph represent?
What about this one?
Introduction
You can solve inequalities involving addition and
subtraction in the same way as you solved
addition and subtraction equations.
You can also graph the solutions on a number
line.
Properties of Inequality
You have solved addition and subtraction
equations by performing inverse operations that
isolate the variable on one side. The value on
the other side is the solution. Inequalities
involving addition and subtraction can be solved
similarly using the following inequality
properties. These properties are also true for ≥
and ≤.
Properties of Inequality
Addition Property of Inequality If 𝑎
If 𝑎
Subtraction Property of
If 𝑎
Inequality
If 𝑎
> 𝑏, then 𝑎 + 𝑐
< 𝑏, then 𝑎 + 𝑐
> 𝑏, then 𝑎 − 𝑐
< 𝑏, then 𝑎 − 𝑐
>𝑏+𝑐
<𝑏+𝑐
>𝑏−𝑐
<𝑏−𝑐
How do the Addition and Subtraction Properties of
Inequality compare to the Addition and Subtraction
Properties of Equality?
________________________________________________
________________________________________________
________________________________________________
________________________________________________
Set Notation for Solution Sets
Most linear inequalities have infinitely many
solutions. When using set notation, it is not
possible to list all the solutions in braces. The
solution 𝑥 ≤ 1 in set notation is 𝑥 𝑥 ≤ 1 . Read
this as “the set of all 𝑥 such that 𝑥 is less than or
equal to 1.”
Graphing the Solution Set of a Linear
Inequality
A number line graph can be used to represent
the solution set of a linear inequality.
Adding to Find the Solution Set
Example 1
Solve. Write the solution using set notation.
Graph your solution.
𝑥−3<2
A.
𝑥 − 3 + ____ < 2 + ____
𝑥 < ____
______________ Property of
Inequality; add______ to both
sides.
Simplify.
Write the solution set using set notation. __________________
Graph the solution set on a number line.
Adding to Find the Solution Set
Example 1
Solve. Write the solution using set notation.
Graph your solution.
𝑥 − 5 ≥ −3
B.
𝑥 − 5 + ____ ≥ −3 + ____
𝑥 ≥ ____
______________ Property of
Inequality; add______ to both
sides.
Simplify.
Write the solution set using set notation. __________________
Graph the solution set on a number line.
Adding to Find the Solution Set
Example 1
1. Is 5 in the solution set of the inequality in Part
A? Explain.
_______________________________________
_______________________________________
2. Suppose the inequality symbol in Part A had
been >. Describe the solution set.
_____________________________________
3. Suppose the inequality symbol in Part B had
been ≤. Describe the solution set.
_______________________________________
Subtracting to Find the Solution Set
Example 2
Solve. Write the solution using set notation.
Graph your solution.
𝑥+4>3
A.
𝑥 + 4 − ____ > 3 − ____
𝑥 > ____
______________ Property of
Inequality
Simplify.
Write the solution set using set notation. __________________
Graph the solution set on a number line.
Subtracting to Find the Solution Set
Example 2
Solve. Write the solution using set notation.
Graph your solution.
𝑥 + 2 ≤ −1
B.
𝑥 + 2 − ____ ≤ −1 − ____
𝑥 ≤ ____
______________ Property of
Inequality
Simplify.
Write the solution set using set notation. __________________
Graph the solution set on a number line.
Adding to Find the Solution Set
Example 2
1. Is −3 in the solution set of the inequality in Part
B? Explain.
_______________________________________
_______________________________________
2. Suppose the inequality symbol in Part A had
been ≥. Describe the solution set.
_____________________________________
3. Suppose the inequality symbol in Part B had
been <. Describe the solution set.
_______________________________________
Summary
1. How can you use properties to justify
solutions to inequalities that involve addition
and subtraction?
2. Describe how to solve an inequality involving
addition and an inequality involving
subtraction. Include the properties that
justify the steps in your description. Write
your solutions in set notation and represent
them with graphs on a number line.
Solving Inequalities by
Multiplying or Dividing
How can you use properties to justify
solutions to inequalities that involve
multiplication and division?
Introduction
Suppose we had to solve the equations
6𝑥 = −42
and
𝑥
3
= 4.
What properties would justify the solution
steps?
You can use a similar process and similar
properties to justify the solutions of inequalities.
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
_________
__________
_________
__________
Both statements are still true: 16 is less than 20,
and 5 is greater than 4.
Multiplying or Dividing by a Negative
Number
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
_________
__________
_________
__________
Is −16 less than −20? Is −5 greater than −4?
Multiplying or Dividing by a Negative
Number
Repeat the multiplication by −4 and the division
by −3, but this time reverse the inequality
symbol when you do.
4<5
15 > 12
_________
__________
_________
__________
Do you get a true statement in each case?
Multiplying or Dividing by a Negative
Number
1. When solving inequalities, if you multiply by
a negative number, you must
_____________________________________
2. When solving inequalities, if you divide by a
negative number, you must
_____________________________________
More Properties of Inequality
You can use the following inequality properties
to solve inequalities involving multiplication and
division. These properties are also true for ≥
and ≤.
Multiplication Property of Inequality
If 𝑎 > 𝑏 and 𝑐
If 𝑎 < 𝑏 and 𝑐
If 𝑎 > 𝑏 and 𝑐
If 𝑎 < 𝑏 and 𝑐
> 0, then 𝑎𝑐
> 0, then 𝑎𝑐
< 0, then 𝑎𝑐
< 0, then 𝑎𝑐
Division Property of Inequality
If 𝑎 > 𝑏 and 𝑐 > 0, then > .
𝑎
𝑐
𝑎
𝑐
𝑎
𝑐
𝑎
𝑐
> 𝑏𝑐.
< 𝑏𝑐.
< 𝑏𝑐.
> 𝑏𝑐.
If 𝑎 < 𝑏 and 𝑐 > 0, then <
If 𝑎 > 𝑏 and 𝑐 < 0, then <
If 𝑎 < 𝑏 and 𝑐 < 0, then >
𝑏
𝑐
𝑏
.
𝑐
𝑏
.
𝑐
𝑏
.
𝑐
Multiplying to Find the Solution Set
Example 1
Solve. Write the solution using set notation.
Graph your solution.
𝑥
A.
>3
2
____
𝑥
2
>____ 3
𝑥 >____
____________ Property of
Inequality
Simplify.
Solution set:____________________
Multiplying to Find the Solution Set
Example 1
Solve. Write the solution using set notation.
Graph your solution.
𝑥
B.
≤ −2
−4
____
𝑥
−4
≥____ −2
𝑥 ≥____
____________ Property of
Inequality; _________ ≤
symbol.
Simplify.
Solution set:____________________
Multiplying to Find the Solution Set
Example 1
1. Suppose the inequality symbol in Part A had
been ≥. Describe the solution set.
_____________________________________
2. Suppose the inequality symbol in Part B had
been <. Describe the solution set.
_____________________________________
Multiplying to Find the Solution Set
Example 2
Solve. Write the solution using set notation.
Graph your solution.
3𝑥 ≥ −9
A.
3𝑥
9
≥−
_____
_____
𝑥 ≥____
____________ Property of
Inequality
Simplify.
Solution set:____________________
Multiplying to Find the Solution Set
Example 2
Solve. Write the solution using set notation.
Graph your solution.
−5𝑥 < 20
B.
−
5𝑥
20
>
______ ______
𝑥 >____
____________ Property of
Inequality; ___________ <
symbol.
Simplify.
Solution set:____________________
Dividing to Find the Solution Set
Example 2
1. There is a negative number in both Parts A and B.
Why is the inequality symbol only reversed in Part B?
___________________________________________
___________________________________________
2. Suppose the inequality symbol in Part A had been >.
Describe the solution set.
___________________________________________
3. Suppose the inequality symbol in Part B had been ≤.
Describe the solution set.
___________________________________________
Summary
Describe how to solve an inequality involving
multiplication and an inequality involving
division. Include the properties that justify your
steps. Write your solutions in set notation and
represent them with graphs on a number line.
Solving Two-Step and Multi-Step
Inequalities
How can you use properties to justify
solutions to multi-step inequalities?
Introduction
Give the steps and properties necessary to solve
the equation
−3𝑥 + 7 = 43
Now, solve the inequality
−3𝑥 + 7 ≤ 43
using your solution to the corresponding
equation.
Solving Inequalities With More Than One Step
Example 1
Find the solution set. Justify each step and
graph the solution set.
4𝑥 − 3 + 𝑥 + 8 > 20
4𝑥 + ___ − ___ + 8 > 20
___ + ___ > 20
5𝑥 + 5 − ___ > 20 − ___
___ > ___
5𝑥 15
>
___ ___
𝑥 > ____
Solution set: __________________
_____________ Property of Addition
Combine like terms
_____________ Property of Inequality
Simplify.
_____________ Property of Inequality
Simplify.
Solving Inequalities With More Than One Step
Example 1
1. How would the solution set change if the
inequality symbol were ≥ rather than >?
_____________________________________
2. How would the solution process be different
if the first term were −4𝑥?
_____________________________________
_____________________________________
The Distributive Property
Occasionally you may need to use the
distributive property to solve an inequality.
Recall that, if 𝑎, 𝑏, and 𝑐 are real numbers, then
𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐.
Using the Distributive Property
Example 2
Find the solution set. Justify each step and
graph the solution set.
−2 3𝑥 − 8 < 10
___ + ___ < 10
−6𝑥 + 16 − ___ < 10 − ___
___ < ___
6𝑥 −6
<
___ ___
𝑥___ ____
Solution set:______________
______________ Property
______________ Property of Inequality
Simplify.
______________ Property of Inequality
______________ the inequality symbol;
simplify.
Using the Distributive Property
Example 2
1. Why was the inequality symbol not reversed
when you multiplied by −2 using the
Distributive Property?
_____________________________________
_____________________________________
_____________________________________
Summary
Create a table that shows the properties of
inequality. The table should give each rule in
words and an example of the rule.
Solving Inequalities with
Variables on Both Sides
How can you use properties to justify
solutions of inequalities with
variables on both sides?
Introduction
What is the goal of the method for solving
2𝑥 + 4 = 𝑥 − 9?
Do you think the Properties of Inequality would
apply to variable terms also? Why?
Using Properties to Justify Solutions
Example 1
Find the solution set. Justify each step and
graph your solution.
3 2𝑥 − 3 ≤ 𝑥 + 1
6𝑥 − 9 ≤ 𝑥 + 1
6𝑥____ − 9 ≤ 𝑥___ + 1
______________ ≤ _________
5𝑥 − 9 + 9 ≤ _______
_______ ≤ ________
______ ≤ ________
_______ ≤ ________
Solution set:____________________
_________________ Property
_________________________________
Simplify.
_________________________________
Simplify.
_________________________________
Simplify.
Using Properties to Justify Solutions
Example 1
1. Why is the Distributive Property applied first in
the solution?
_______________________________________
2. Could the properties have been applied in a
different order than shown in Example 1? If so,
would this make finding the solution easier or
more difficult? Explain.
_______________________________________
_______________________________________
_______________________________________
Using Properties to Justify Solutions
Example 1
3. How would the solution change if the
simplified coefficient of 𝑥 were negative?
_____________________________________
_____________________________________
Using Properties to Justify Solutions
Example 2
Find the solution set. Justify each step and
graph your solution.
2𝑥 − 3 ≤ 3(4 − 𝑥)
Summary
Write a step-by-step method for solving
inequalities with variables on both sides.
One method should involve the distributive
property.
Solving Compound Inequalities
How can you solve special compound
inequalities?
Introduction
Recall that a set is a collection of objects.
Let 𝐴 and 𝐵 be two sets. Then, the union of 𝐴
and 𝐵, denoted 𝐴 𝐵, is the set containing all
of the elements in either 𝐴 or 𝐵.
The intersection of 𝐴 and 𝐵, denoted 𝐴 𝐵, is
the set containing all of the elements in both 𝐴
and 𝐵.
Compound Inequalities
Compound inequalities are two inequalities joined
by AND ( ) or OR ( ).
To solve a compound inequality:
1. Solve each inequality independently.
2. Graph the solutions above the same number
line.
3. Decide which parts of the graphs represent the
solution. If AND is used, it’s the common points.
If OR is used, it’s all points. Then graph the
solution on the number line.
Solving Compound Inequalities
Example 1
Solve. Write the solution in set notation. Graph
the solution.
2𝑥 < 8 AND 3𝑥 + 2 > −4
𝑥<4
3𝑥 > −6
𝑥 > −2
Solution set: 𝑥 𝑥 < ____
or
𝑥 ____< 𝑥 < ____
𝑥 𝑥 > ____
Solving Compound Inequalities
Example 2
Solve. Write the solution in set notation. Graph
the solution.
3𝑥 + 2 ≥ −1 OR 4 − 𝑥 ≥ 2
3𝑥 ≥ −3
−𝑥 ≥ −2
𝑥 ≥ −1
𝑥 ≤ −2
Solution set: 𝑥 𝑥 ≥
𝑥𝑥≤
Or
The set of all ___________numbers
Solving Compound Inequalities
Example 3
Solve. Write the solution in set notation. Graph
the solution.
2𝑥 − 3 > 3 AND 𝑥 + 4 ≤ 1
2𝑥 > 6
𝑥 ≤ −3
𝑥>3
Solution set: 𝑥 𝑥 ≤ ____
𝑥 𝑥 > ____
or the ___________ set or _____
Solving Compound Inequalities
Example 4
Solve. Write the solution in set notation. Graph
the solution.
3𝑥 − 1 > 2 OR 2𝑥 + 2 ≥ 8
3𝑥 > 3
2𝑥 ≥ 6
𝑥>1
𝑥≥3
Solution set: 𝑥 𝑥 >
Or
𝑥𝑥>
𝑥𝑥≥
Solving Compound Inequalities
1. In Example 3, why is the solution set the
empty set?
_____________________________________
_____________________________________
2. In Example 4, why is the solution set
𝑥𝑥>1 ?
_____________________________________
_____________________________________
Summary
In your own words, describe how to solve
compound inequalities.
Solving Absolute Value Equations
How can you use graphing to solve
equations involving absolute value?
Introduction
Suppose we had to solve the equation
3
𝑥
2
+ 1 = 4.
3
𝑥
2
One approach would be to consider 𝑦 =
+1
and 𝑦 = 4 as individual equations. We could
then graph each equation. What would the
intersection of the graphs tell us?
Solve the equation algebraically as well.
Solving an Absolute Value Equation by Graphing
Example 1
Use a graphing calculator to solve the absolute value equation 𝟐 𝒙 − 𝟑 + 𝟏 = 𝟓.
1.
2.
3.
Treat the left side of the
equation as the absolute value
equation 𝑦 = 2 𝑥 − 3 + 1 and
the right side as 𝑦 = 5.
Graph both equations using
your calculator.
Identify the 𝑥-coordinate of
each point where the graphs
intersect. Show that each 𝑥coordinate is a solution of
2 𝑥 − 3 + 1 = 5.
__________________________
__________________________
Solving an Absolute Value Equation by Graphing
Example 1
1. Why is the 𝑦-coordinate of both points of
intersection equal to 5?
_____________________________________
2. The vertex of an absolute value graph is the
lowest point if the graph opens upward or the
highest point if the graph opens downward. The
vertex of the graph of 𝑦 = 2 𝑥 − 3 + 1 is 1,3 .
How are the coordinates of the vertex related to
its equation?
_______________________________________
_______________________________________
Solving an Absolute Value Equation Using Algebra
Example 2
Solve the equation 𝟐 𝒙 − 𝟑 + 𝟏 = 𝟓 using
algebra.
A. Isolate the expression 𝑥 − 3 .
2 𝑥−3 +1=5
Write the equation.
2 𝑥 − 3 + 1 − ____ = 5 − ____ Subtract 1 from both sides.
2 𝑥 − 3 = ____
Simplify.
2 𝑥−3
Divide both sides by 2.
=
𝑥 − 3 = _____
Simplify
Solving an Absolute Value Equation Using Algebra
Example 2
Solve the equation 𝟐 𝒙 − 𝟑 + 𝟏 = 𝟓 using
algebra.
B. Interpret the equation 𝑥 − 3 = 2: What
numbers have an absolute value equal to 2?
_______________________________________
C. Set the expression inside the absolute value bars
equal to each of the numbers from Part B and solve
for 𝑥.
𝑥 − 3 = ____
or
𝑥 − 3 + ___ = ___ + ___
𝑥 = ____
or
𝑥 − 3 = ____
Write an equation for
each value of 𝑥 − 3
𝑥 − 3 + ___ = ___ + ___
Add 3 to both sides of
both equations.
𝑥 = ____
Simplify.
Solving an Absolute Value Equation Using Algebra
Example 2
1. The left side of the equation is 2 𝑥 − 3 + 1. Evaluate
this expression for each solution of the equation. How
does this help you check the solutions?
___________________________________________
___________________________________________
2. Suppose the number on the right side of the equation
was −5 instead of 5. What solutions would the
equation have? Why? When answering these
questions, you may want to refer to the graph of 𝑦 =
2 𝑥 − 3 + 1.
___________________________________________
___________________________________________
Solving a Real-World Problem
Example 3
Sal exercises by running 3 miles along a road in
front of his house and then reversing his
direction to return home. He runs at a constant
speed of 0.1 mile per minute. Write and graph
an absolute value equation that gives his
distance 𝑑 (in miles) from home in terms of the
elapsed time 𝑡 (in minutes). Use the graph to
find the time(s) at which Sal is 1 mile from
home.
Solving a Real-World Problem
Example 3
A. Determine the three key values of the distance
equation:
• When Sal begins his run (𝑡 = 0 minutes), he is ____ miles
from home, so 𝑑 =_____.
• When Sal reverses direction, he is _____ miles from home.
______ miles
He reaches this point in 𝑡 =
minutes,
0.1 mile per minute
so when 𝑡 =_____, 𝑑 =______.
• When Sal returns home, he is ____ miles from home.
Because he has run a total of 6 miles, he reaches this point
______ miles
in 𝑡 =
=____ minutes, so when
0.1 mile per minute
𝑡 =____, 𝑑 =_____.
Solving a Real-World Problem
Example 3
B. Add axis labels and scales to
the coordinate plane shown, then
plot the points 𝑡, 𝑑 using the
time and distance values from
Part A. The equation is an
absolute-value equation, and the
vertex of the equation’s graph is
the point that represents when
Sal reverses direction. Draw the
complete graph and then write
the absolute value equation.
𝑑 = −0.1 𝑡 − ___ + ___
C. To find the time(s) when Sal is
1 mile from home, draw the
graph of 𝑦 = 1. Find the 𝑡coordinate of each point where
the two graphs intersect.
___________________________
Solving a Real-World Problem
Example 3
1. Show how to use algebra to find the time(s)
when Sal is 1 mile from home.
Summary
Describe how you would solve an equation of
the form
𝑎 𝑥−ℎ +𝑘 =𝑐
by graphing.
Solving Absolute Value
Inequalities
How does solving absolute value
inequalities relate to solving
compound inequalities?
Introduction
Recall that the absolute value of a number is its
distance from 0.
What would the solution of 𝑥 = 4 look like on a
number line?
What would the solution of 𝑥 > 4 look like on a
number line?
What would the solution of 𝑥 < 4 look like on a
number line?
Solving Absolute Value Inequalities with <
Example 1
Solve the inequality 𝑥 + 2 < 5.
A. Use the Subtraction Property of Inequality to isolate the absolute
value expression.
𝑥 +2<5
𝑥 + 2 −___< 5 −___
𝑥 <___
B. Write a description of the solution of the inequality.
All real numbers that are _____ than _____ units from 0
C. Draw the graph of the solution on the number line.
D. Use the graph to rewrite 𝑥 < 3 as a compound inequality.
𝑥 >____ _________ 𝑥 <___, or ___< 𝑥 <___
Solving Absolute Value Inequalities with <
Example 1
1. How did you decide whether to use AND or
OR in the compound inequality?
_____________________________________
_____________________________________
2. Solve 𝑥 + 2 ≤ 5 by rewriting it as a
compound inequality. Show your work.
_____________________________________
_____________________________________
_____________________________________
Solving Absolute Value Inequalities with >
Example 2
Solve the inequality 𝑥 − 3 > 2.
A. Use the Addition Property of Inequality to isolate the absolute
value expression.
𝑥 −3>2
𝑥 − 3 +___> 2 +___
𝑥 >___
B. Write a description of the solution of the inequality.
All real numbers ___________ than _____ units from 0.
C. Draw the graph of the solution on the number line.
D. Use the graph to rewrite 𝑥 > 5 as a compound inequality.
𝑥 <___ ________ 𝑥 >___
Solving Absolute Value Inequalities with >
Example 2
1.
2.
3.
How did you decide whether to use AND or OR in the compound
inequality?
_____________________________________________________
_____________________________________________________
Solve 𝑥 − 3 ≥ 2 by rewriting it as a compound inequality. Show
your work.
_____________________________________________________
_____________________________________________________
_____________________________________________________
Write a generalization for absolute value inequalities that relate
the inequality symbol to the type of compound inequality that
represents a solution.
_____________________________________________________
_____________________________________________________
_____________________________________________________
Summary
Describe how to solve an absolute value
inequality. Address how you know whether the
solution of the inequality will result in an AND or
an OR compound inequality.