6. Solving Inequalities
+ Customize lesson
0
0
Sign In
An error occurred, try again later!
Chapter 6
6.
Solving Inequalities
This lesson delves into the methods for solving inequalities, focusing on the properties of inequalities, real numbers, and boundary points. It explains how these properties can be used to isolate variables and find the solution set. For example, the Addition and Subtraction Properties of Inequalities are discussed as tools for simplifying inequalities. The lesson also provides practical scenarios, such as determining the number of seats in a theater or the height requirement for joining a club, to illustrate the application of these mathematical concepts. It emphasizes that inequalities can have multiple or even infinitely many solutions, which can be represented on a number line.
| Student Learning Objectives: |
|---|
|
| | 15 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Solving Inequalities
Slide of 15
Inequalities can be solved like equations. The key step is to isolate the variable on one side of the inequality. In this lesson, several examples are introduced showing how to use the properties of inequalities to solve them.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Share
Report error
Translate
Challenge
Tickets for a Drama: Making Money
Jordan is a member of the drama club at her school. The club plans to stage the play The Little Frog in Town this weekend. They charge $6 per ticket.
a
The club hopes to earn at least $400 from the play. If the club has already sold 42 tickets, write an inequality to represent the number of tickets they still need to sell.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["252+6x \\geq 400","6x \\geq 148","x \\geq 25 ","400 \\leq 252+6x","148 \\leq 6x","x\\leq 25"]}}
b
What is the minimum total number of tickets the club needs to sell to reach their goal?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":"tickets","answer":{"text":["67"]}}
Share
Report error
Translate
Discussion
Properties of Inequalities
Not all inequalities start in the form x< a, x > a, x ≤ a, or x ≥ a. We can use inverse operations and the Properties of Inequalities to rewrite them into one of these forms. Consider the Addition and Subtraction Properties of Inequalities.
Rule
Addition Property of Inequality
Adding the same number to both sides of an inequality results in an equivalent inequality. The solution set and inequality sign of this equivalent inequality does not change. Let x, y, and z be real numbers such that x< y. Then, the following conditional statement holds true.
If x< y, then x+z< y+z.
This property holds for the other types of inequalities.
| Addition Property of Inequality | |
|---|---|
| x< y | x+ z< y+ z |
| x> y | x+ z> y+ z |
| x≤ y | x+ z≤ y+ z |
| x≥ y | x+ z≥ y+ z |
Share
Report error
Translate
Discussion
Subtraction Property of Inequality
Subtracting the same number from both sides of an inequality results in an equivalent inequality. The solution set and inequality sign of this equivalent inequality does not change. Let x, y, and z be real numbers such that x< y. Then, the following conditional statement holds true.
If x< y, then x-z< y-z.
This property holds for the other types of inequalities.
| Subtraction Property of Inequality | |
|---|---|
| x< y | x- z< y- z |
| x> y | x- z> y- z |
| x≤ y | x- z≤ y- z |
| x≥ y | x- z≥ y- z |
Share
Report error
Translate
Discussion
Multiplication Property of Inequality
Multiplying both sides of an inequality by a nonzero real number z results in an equivalent inequality. The following conditions about z need to be considered when applying this property.
| Positive z | If z is positive, the inequality sign remains the same. |
|---|---|
| Negative z | If z is negative, the inequality sign reverses to produce an equivalent inequality. |
Let x, y, and z be real numbers such that x< y and z≠0. Then, the equivalent inequalities can be written depending on the sign of z.
- If x< y and z> 0, then xz< yz.
- If x< y and z< 0, then xz> yz.
This property holds for the other types of inequalities.
Share
Report error
Translate
Discussion
Division Property of Inequality
Dividing both sides of an inequality by a nonzero real number z results in an equivalent inequality. The following conditions about z need to be considered when applying this property.
| Positive z | If z is positive, the inequality sign remains the same. |
|---|---|
| Negative z | If z is negative, the inequality sign reverses to produce an equivalent inequality. |
Let x, y, and z be real numbers such that x
- If x
0, then xz< yz. - If x
yz.
This property holds for the other types of inequalities.
Share
Report error
Translate
Pop Quiz
Identifying the Appropriate Property to Solve the Inequality
Determine which property to use to isolate the variable on one side of the inequality to produce the given equivalent inequality.
Share
Report error
Translate
Discussion
Equivalent Inequalities
Two or more inequalities are equivalent inequalities if they have the same solution set. Similar to equivalent equations, applying Properties of Inequality to an inequality produces an equivalent inequality. Consider the following example. 3x+21>6-2x Three Properties of Inequalities will be used to solve this inequality.
The solution set of every inequality created in this process is x>-3. The Properties of Inequalities produced three equivalent inequalities until the solution set was found.
Share
Report error
Translate
Example
Ordering Merch
A school is ordering T-shirts and hats to sell as a fundraiser for the baseball team.
The number of shirts on order s is described by the inequality.
4s-132 ≤ s+174
The number of hats on order h is defined by another inequality.
143-3h > h-17
a
Find the simplest form of the inequality for the number of shirts on order.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["s"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["s \\leq 102","102 \\geq s"]}}
b
Find the simplest form of the inequality for the number of hats on order.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["h"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["40 \\gt h","h \\lt 40"]}}
c
What is the maximum number of merch items on the order according to the given inequalities?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":"items","answer":{"text":["141"]}}
Hint
a
Use the Properties of Inequalities to simplify the given inequality.
b
Use the Properties of Inequalities to simplify the given inequality.
c
Use the simplified inequalities from Parts A and B.
Solution
a
We will start by examining the given inequality that represents the number of shirts on order.
4s-132 ≤ s+174 Let's use the Properties of Inequalities to simplify this inequality. Since the variable s appears on both sides of the inequality, we will subtract s from both sides of the inequality using the Subtraction Property of Inequality.
Next, let's add 132 to both sides of the inequality using the Addition Property of Inequality.
Finally, we will divide both sides of the inequality by 3 to isolate the variable using the Division Property of Inequality.
We have found the simplest form of the given inequality. This means that the number of shirts on the order is less than or equal to 102. In other words, there are at most 102 shirts on order.
b
Now let's consider the second inequality to simplify.
143-3h > h-17 We will use the Properties of Inequalities to simplify it. Let's start by adding 3h to both sides of the inequality by using the Addition Property of Inequality.
Now, we will add 17 to both sides of the inequality.
As the last step, let's divide both sides of the inequality by 4. Since 4 is a positive number, the inequality sign stays the same.
The simplest form of the given inequality is found. 143-3h > h-17 ⇕ 40 > h or h < 40 This means that 40 is greater than the number of hats on the order. In other words, the school is ordering at most 39 hats.
c
Recall the simplified forms of the inequalities in Parts A and B.
ccc Shirts & and & Hats s ≤ 102 & & h < 40 According to these inequalities, there are at most 102 shirts and and 39 hats on the order. Now, let's add these numbers to find the maximum number of merch items on order. 102+39=141 The maximum number of items on the order is 141.
Share
Report error
Translate
Discussion
Solution Set of an Inequality
A solution of an inequality is any value of the variable that makes the inequality true. If a value is substituted for the variable and creates a false statement, that value is not in a solution of the inequality. Consider the following inequality when x=-3, 0, and 3. 2x-3< 5 lcl 2( -3)-3? <5 & ⇒ & -9 < 5 ✓ [0.3em] 2( 0)-3? <5 & ⇒ & -3 < 5 ✓ [0.3em] 2( 3)-3? <5 & ⇒ & 6 ≮ 5 * The set of all possible values that satisfy an inequality is the solution set of an inequality. The solution set of the inequality can be represented using set-builder notation. \begin{gathered} \underline\textbf{Solution Set} \\ \{x\,|\,x\lt 4\} \end{gathered} The solution set of a linear inequality in one variable can also be represented using a number line.
Share
Report error
Translate
Example
A Day at the Amusement Park
Tadeo and his friends are spending the day at an amusement park.
a
One roller coaster has a height requirement. A rider's height h in centimeters must satisfy the following inequality.
-2h+30 ≤ h-315 Find the solution set of the inequality for the minimum height of a rider.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["h"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"{h |","formTextAfter":"}","answer":{"text":["h \\geq 115","115 \\leq h"]}}
b
Tadeo tries his hand at the ring toss game on the midway. Players earn points by tossing rings onto bottles. To win the biggest prize, their score s must satisfy the inequality.
5(s-14) > 245 Find the solution set of the inequality. What score does Tadeo need to get to win the biggest prize?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["s"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"{s |","formTextAfter":"}","answer":{"text":["s \\gt 63","63 \\lt s"]}}
Hint
a
Isolate the variable h by using the Properties of Inequalities.
b
Isolate the variable s by using the Properties of Inequalities.
Solution
a
Let's simplify the given inequality using the Properties of Inequalities to isolate the variable h.
-2h+30 ≤ h-315 First, we will subtract 30 from both sides of the inequality using the Subtraction Property of Inequality.
Next, let's subtract h from both sides of the inequality.
We will now divide both sides of the inequality by -3 to isolate the variable h. Note that by the Division Property of Inequality, the inequality sign is reversed since the divisor -3 is a negative number.
-3h≤ -345
DivNegIneq
Divide by -3 and flip inequality sign
-3h/-3 ≥ -345/-3
CalcQuot
Calculate quotient
h≥ 115
Finally, let's write the simplified inequality in the set-builder notation notation. h≥ 115 ⇒ {h | h ≥ 115 } The minimum height of riders for the roller coaster is 115 centimeters.
b
We want to find the solution set of the following inequality.
5(s-14) > 245 We will proceed in the same way as we did in Part A. Let's isolate the s-variable expression by dividing both sides of the inequality by 5.
5(s-14) > 245
DivIneq
.LHS /5.>.RHS /5.
5(s-14)/5 > 245/5
CancelCommonFac
Cancel out common factors
5(s-14)/5 > 245/5
CalcQuot
Calculate quotient
s-14 > 49
Next, we will add 14 to both sides of the inequality to isolate the variable s.
Finally, let's write the simplified inequality in the set-builder notation. s > 63 ⇒ {s | s > 63 } Tadeo needs to get more than 63 points at the ring toss game to win the biggest prize.
Share
Report error
Translate
Pop Quiz
Solving Inequalities
Select the correct solution set for the given inequality.
Share
Report error
Translate
Discussion
Graphing an Inequality on a Number Line
A number line can be used to represent the solution set of an inequality in one variable. Consider the following inequality. x+2<8 There are four steps for graphing the given inequality.
1
Determine the Type of Inequality
expand_more The first step is to determine whether the inequality is strict or non-strict.The given inequality is strict because the inequality symbol is <. x + 2 < 8_(Strict)
2
Determine the Solution Set and the Boundary Point
expand_more Next, the solution set and the boundary point of the inequality need to be found by solving the inequality using the Properties of Inequalities.
The boundary point is 6 and the solution set corresponds to all real numbers less than 6.
3
Draw the Boundary Point on the Number Line
expand_more Draw a circle representing the boundary point on the number line. If the inequality is strict, the circle is open (∘). If the inequality is non-strict, the circle is closed (∙). The inequality is strict for this example, so draw an open circle on the boundary point 6.
4
Shade the Rest of the Solution Set
expand_more Finally, shade the rest of the solution set by drawing an arrow that starts on the boundary point and goes along the solution set. The solution set of the given inequality is all numbers less than 6, which means that the arrow will point to the left of the boundary point.
The graph of inequalities whose solution sets are all the real numbers is represented with bidirectional arrows that cover all the number line.
Share
Report error
Translate
Example
Planning a Museum Tour
A museum asks visitors for feedback about its guided tours. Many visitors say the tours are too long.
Some tours currently last up to 2 hours. The museum decides that future guided tours should last less than 90 minutes.
a
Write an inequality for the duration d in minutes of guided tours going forward.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["d"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["d<90","90>d"]}}
b
Choose the graph that describes the solution set of the inequality written in Part A.
{"type":"choice","form":{"alts":["Graph I","Graph II","Graph III","Graph IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
a
Use a strict inequality to represent the phrase less than.
b
The solution set includes real numbers, not only integers.
Solution
a
The museum directors decide to shorten the length of the guided tours to less than 90 minutes. If d represents the duration of the tours in minutes, the following inequality represents the directors' decision.
d < 90 We can also write the inequality using the phrase greater than. In other words, saying d is less than 90 is the same as saying that 90 is greater than the duration of the tour d. d < 90 ⇔ 90 > d
b
Let's sketch the graph of the inequality we wrote in Part A.
d < 90 Its solution set consists of all real numbers that are smaller than 90. We can show it on the number line as an arrow from 90 and pointing to the left. Since the inequality is strict, 90 is not in the solution set, which we will mark it with an open circle.
The above sketch matches Graph I.
Share
Report error
Translate
Closure
Finding the Number of Tickets
At the beginning of the lesson, we were told that the drama club charges $6 per ticket to a play and that 42 tickets to the show have been already sold. First, let's calculate the amount of money earned from these 42 tickets. 42 * 6 =252 The club earned $252 from selling 42 tickets. They want to earn at least $400 from this play, so $400 should be less than or equal to their total earnings. We can write this fact as the following inequality. 400 ≤ Total Earnings Let x be the number of additional tickets the club needs to sell after selling 42 tickets. The club will earn 6x dollars by selling them and then the club's total earnings will equal 252+6x dollars. Let's complete our inequality using this fact. 400 ≤ 252+6x We will now solve this inequality to find the minimum number of tickets sold. We will use the Properties of Inequalities to isolate the variable x. Let's start by subtracting 252 from both sides of the inequality using the Subtraction Property.
We will then divide both sides of the inequality by 6 using the Division Property.148 ≤ 6x
DivIneq
.LHS /6.≤.RHS /6.
148/6 ≤ 6x/6
CalcQuot
Calculate quotient
24.666666 ... ≤ x
RoundDec
Round to 2 decimal place(s)
24.67 ≤ x
Share
Report error
Translate
Solving Inequalities
Exercise 2.1
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["1.15x \\gt 45","45 \\lt 1.15x"]}}
{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
security
report
code
security
report
code
We want to write an inequality that represents how many times Izabella can ride the subway in a month so that a monthly pass is a better deal. We know that a single subway ride costs $ 1.15 and a monthly pass costs $ 45.
| Subway Rides Costs | |
|---|---|
| Single-ride Ticket | $ 1.15 |
| Monthly Pass | $ 45 |
Let x be the number of times Izabella rides the subway in a month. We can find the price of x subway rides on single-ride tickets by multiplying x by the price of a single pass. 1.15x Izabella can also buy a monthly pass and pay $ 45 for x rides. If a monthly pass is a better deal, then it is cheaper than 1.15x. This means that 1.15x is more than 45. Let's write it as an inequality. 1.15x > 45
It is a given that Izabella rides a subway about 50 times per month. In Part A we found the inequality that represents how many times Izabella can ride the subway so that a monthly pass is a better deal than buying single-ride tickets.
1.15x > 45
In the inequality the variable x is the number of times Izabella rides the subway in a month. We will substitute 50 for x and check whether the inequality produces a true statement.
The inequality produces a true statement for x = 50. This means that buying a monthly pass is a better deal than buying single-ride tickets.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["x+32 \\leq 55","x \\leq 23"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["x+27.2 \\leq 55","x \\leq 27.8"]}}
security
report
code
security
report
code
Izabella can spend up to $ 55 and she wants to buy a dress that costs $ 32. Let x be the remaining amount of money that she can spend after buying the dress. We will write and solve an inequality to find the possible values of x. The total amount of money that she can spend is the sum of the dress price and x. 32 + x We know that she can spend up to $ 55. This means that the total amount of money that she can spend is less than or equal to 55. Let's write it as a non-strict inequality. 32 + x ≤ 55 Next, we will subtract 32 from each side of the inequality to isolate the variable x by using the Subtraction Property of Inequality.
This means that Izabella can spend at most $23. The remaining amount of money after buying the dress can be shown by the following inequalities. 32 + x ≤ 55 ⇕ x ≤ 23
The dress that costs $ 32 is on sale for 15 %. We will find the new price of the dress and rewrite the inequality from Part A. First, let's find the amount of discount by calculating 15 % of 32.
The new price of the dress is the original price minus the amount of the discount. 32- 4.8 = 27.2 We found that the price of the dress on sale is $ 27.20. In Part A we wrote the inequality when the price of the dress is $ 32. The inequality can be used to find the amount of money that Izabella can spend, x. Let's consider the inequality from Part A. 32 + x ≤ 55 The new price of the shirt is $ 27.20. We will write the same inequality but instead of 32, we will write 27.2. 27.2 + x ≤ 55 Notice that we can rearrange the inequality by isolating the variable to solve it. Let's subtract 27.2 from both sides of the inequality.
This means that Izabella can spend at most $27.80 after buying the dress on sale. 27.2 + x ≤ 55 ⇕ x ≤ 27.8
security
report
code
E
Hint & Answer
S
Solution
Students in a science class are divided into 4 equal groups for a project. The teacher says each group should include at least 5 students. Which of the numbers are the possible number of students in the class?
{"type":"multichoice","form":{"alts":["18","16","20","24","30"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":[2,3]}
security
report
code
security
report
code
Students in a science class are divided into 4 equal groups. Let n be the number of students in the science class. We can find the number of students in each group by dividing the total number of students by the number of groups. n/4 We know that in each group there are at least 5 students. This means that n4 should be greater than or equal to 5. Let's write it as an inequality. n/4 ≥ 5 We will solve the inequality to find the possible values of n. Let's multiply both sides of the inequality by 4 by using the Multiplication Property of Inequality.
This means that the number of students should be greater than equal to 20. Notice that the number of students should be a natural number. Let's compare the given numbers with 20. 16 < 20 * 18 < 20 * 20 = 20 ✓ 24 ≥ 20 ✓ 30 ≥ 20 ✓ It was also said that the groups are equal which means they should include equal number of students. Since there are 4 groups, we should consider the numbers that are divisible by 4. With this in mind, let's consider the numbers greater than equal to 20 once again. 20 ÷ 4 =5 ✓ 24 ÷ 4=6 ✓ 30 ÷ 4=7.5 * Since the number of students should be a natural number, there cannot be 7.5 students in a group. The possible numbers are 20 and 24.
security
report
code
E
Hint & Answer
S
Solution
Solving Inequalities
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
Edit Lesson
≥
>
■2
■▢
e▢
7
8
9
×
÷■1
=
=
√▢
▢√
∫▢▢
4
5
6
+
−
≤
<
log
ln
log▢
1
2
3
()
∣
sin
cos
tan
0
.
π
x
y
Loading content