12. Solving and Graphing One Variable Compound Inequalities
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Chapter 1
12.
Solving and Graphing One Variable Compound Inequalities
Compound inequalities are mathematical expressions that involve two or more inequalities connected by the words "and" or "or." The content focuses on solving these inequalities using various methods and graphing the solutions. It also highlights the properties of inequalities that are essential for solving them correctly. Real-world scenarios, such as determining safe driving speeds or calculating the range of possible temperatures, are used to illustrate the practical applications of compound inequalities. Understanding these concepts can help in various fields like engineering, economics, and natural sciences.
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| | 11 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Solving and Graphing One Variable Compound Inequalities
Slide of 11
This lesson will work with compound inequalities as well as how to solve them and interpret the solutions.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Going Home from the Beach
Vincenzo is getting ready to drive home from vacation at the beach. He is sure that if he drives at 50 miles per hour, he will be home in 40 to 55 minutes.
a How far can Vincenzo's house be from the beach?
b Supposing Vincenzo's assumption is correct, let y be an invalid distance from his house to the beach. How could the possible values of y be written?
Explore
Combining Inequalities
Consider the following inequalities.
Similarly, to solve the second inequality, 5 needs to be added to both sides of the inequality.
Therefore, the solution set of this inequality are the numbers to the right of 1 in a number line, not including 1. 
What would happen if these two inequalities were combined? For combined inequalities, x might solve both inequalities at the same time or either inequality at any given time. How would these solution sets look? These kinds of combined inequalities arise in all kinds of situations. Because of this, it is important to study them.
1:2x+1<52:5x−5>0
To solve each inequality for x, the Properties of Inequalities need to be used. To solve inequality number one, 1 should be subtracted from both sides first.
The solution set of this inequality can be graphed by marking every number to the left of 2, not including 2. Discussion
Compound Inequalities
Combining two or more inequalities with the word and
or or
yields what is called a compound inequality.
| Compound Inequality | Is Read As |
|---|---|
| x<5 or x>8 | x is less than 5 or greater than 8. |
| x>2 and x≤4 | x is greater than 2 and less than or equal to 4. |
andare commonly written without using this word. Consider the following example.
x>2 and x≤4
The first inequality can be rewritten as 2<x because the statement x is greater than 2is equivalent to
2 is less than x.With this change, the inequality can be rewritten as follows.
2<x and x≤4⇒2<x≤4
It is important to note that compound inequalities written with the variable in the middle, similar to the example above, are always compound inequalities written with and.
Method
Solving a Compound Inequality
Solving a compound inequality is done by first separating each individual inequality. Then, each inequality is solved one at a time in the regular fashion. Lastly, the solution sets are combined. As an example, consider the following compound inequality.
-3<2x−1≤2
This inequality will be solved to illustrate each step of this method.
1
Separate the Compound Inequality
expand_more Before a compound inequality can be solved, each individual inequality has to be identified.
-3<2x−1≤2⇓-3<2x−1 and 2x−1≤2
2
Solve the Individual Inequalities
expand_more The individual inequalities can now be solved one at a time. For the example used, -3<2x−1 will be solved first, using inverse operations.
The solution set is -1<x. Next is 2x−1≤2, which is solved in a similar manner.
The solution set of the second inequality is all values of x that satisfy x≤23.
-3<2x−1
AddIneq
LHS+1<RHS+1
-3+1<2x−1+1
AddTerms
Add terms
-2<2x
DivIneq
LHS/2<RHS/2
2-2<22x
SimpQuot
Simplify quotient
-1<x
2x−1≤2
AddIneq
LHS+1≤RHS+1
2x−1+1≤2+1
AddTerms
Add terms
2x≤3
DivIneq
LHS/2≤RHS/2
22x≤23
SimpQuot
Simplify quotient
x≤23
3
Combine the Solution Sets
expand_more Lastly, combining the two solution sets yields the solution set of the compound inequality. Here, the compound inequality was written in the condensed form of an
andinequality. Therefore, the two solution sets can written separately with the word
and,or they can be combined as follows.
-1<x and x≤23⇓-1<x≤23
Concept
Solution Set of a Compound Inequality
The solution set of a compound inequality consists of the solution sets of the individual inequalities. For compound inequalities with 
A solution of a compound inequality with 
Unlike the graphs of compound inequalities with
or,a solution of either individual inequality is a solution of the compound inequality. Consider the following compound inequality.
x≥2 or x<-2
The graph of this compound inequality is the union of the graphs of the individual inequalities. These graphs are recognized by the fact that they continue infinitely in either direction.
and,however, must be a solution of both individual inequalities. Consider the following inequality.
x<1 and x≥-3
The graph of the compound inequality is the intersection of the graphs of the individual inequalities. or,compound inequalities written with
anddo not always extend infinitely.
Example
The Speed of a Tractor
Kevin's class is having a work experience internship this week. He is working with a farmer to test the speed of an autonomous tractor in a field.
In the first test, the tractor started from a point 2 miles away from the barn. After a half hour, the tractor was at least 10 miles away from the barn. In the next test, the tractor started at a point 1.5 miles away from the barn. When it stopped 45 minutes later, it was less than 20 miles from the barn.
a Considering the test results, what are the possible speeds in miles per hour at which the tractor could drive? Round the answer to one decimal place.
b Graph the possible speeds on a number line.
Answer
a 16≤r<24.7
b
Hint
a Write the given information as inequalities.
b Graph the individual inequalities, then combine the graphs.
Solution
a To find the possible speeds, both test needs to be considered. It is given that in the first test, the tractor started 2 miles away from the barn and stopped at least 10 miles away from the barn after half an hour. This means that the sum of the distance traveled and the starting distance of 2 miles has to be greater than or equal to 10 miles.
distance traveled+2≥10
To find the distance traveled, a modification of the speed formula will be used. r=td
In this formula, r is the speed or rate, d is the distance traveled, and t is the time it took to travel the distance. To find the distance, d needs to be the isolated variable.
r=td⇒d=rt
Therefore, this formula can be substituted for the distance traveled in the written inequality.
rt+2≥10
Since the speed is asked in miles per hour and the tractor drove for half an hour, the given time is written as 21=0.5.
0.5r+2≥10
Likewise in the second test, it is given that the tractor starts from 1.5 miles away and ends less than 20 miles away from the barn after 45 minutes. This means that the sum of the distance traveled and 1.5 miles has to be less than 20 miles.
distance traveled+1.5<20
Then, the formula for the distance traveled can be substituted in this inequality as well.
rt+1.5<20
In this case 45 minutes can be converted to hours using a conversion factor. 45 minutes are equal to 6045=0.75 hours.
0.75r+1.5<20
Since the possible values for the speed are the values of r that satisfy both inequalities, these inequalities can be combined as a compound inequality with and.
0.5r+2≥10 and 0.75r+1.5<20
To solve this compound inequality, the properties of inequalities will be used. To solve the inequality on the left, first 2 will be subtracted from both sides of the inequality.
Similarly, to solve the inequality on the right, 1.5 will be subtracted from both sides of the inequality.
0.75r+1.5<20
r<24.7
r≥16 and r<24.7
It should be noted that some compound inequalities that are written with andcan be rewritten as follows.
16≤r<24.7
This means that the possible speeds of the tractor are greater than or equal to 16 miles per hour and less than 24.7 miles per hour.
b To graph the solution set of the compound inequality, it is useful to graph the solution set of each individual inequality first. Since the inequality is non-strict, the solution set of the inequality r≥16 is made of all the numbers to the right of 16, including 16. This is shown by using a closed circle at 16.
On this case, since the inequality is strict, the solution set of the inequality r<24.7 is made of the numbers to the left of 24.7. Since 24.7 is not included in the solution set, an open circle is used instead.
Since the compound inequality is written with and,
its solution set is made of the numbers that satisfy both inequalities.
Example
Securing the Area Around a Volcano
Ignacio is interning at the local disaster preparedness center. He is using a simulator to help to secure the unsafe area near a volcano that is about to erupt. He is in charge of marking safe distances to the east and the west of the volcano. He initially marked the safe distance with flags, one 30 miles to the east of the base of the volcano and the other 15 miles to the west.
As time passes and data comes in, Ignacio realizes that his estimates were wrong. He notes that the flag to the west is less than half the distance away from the volcano that it should be. On the other hand, he calculates that the eastern flag covers less than two thirds of the actual necessary safe distance from the volcano.
a Consider that the distance to the east of the volcano are represented by positive numbers and that distances to the west are represented by negative numbers. Write a compound inequality that describes the safe distances from the volcano.
b Graph these lengths on a number line.
Answer
a d<-30 or d>45
b
Hint
a Rewrite the given information as individual inequalities.
b Graph the individual inequalities, then combine the graphs.
Solution
a To write the safe distance as a compound inequality, the individual inequalities must first be identified. It is given that the western flag is less than half the actual necessary safe distance. This can be written as an inequality, with d representing the total safe distance.
15 miles west<21d
Since the distance is to the west of the volcano, it is written as a negative number. Also, since this value is negative, further distances from the volcano are less than -15. Therefore, the inequality has to be rewritten as follows.
21d<-15
Similarly, it is given that the eastern flag is less than two thirds of the safe distance at 30 miles from the volcano. This can also be written as an inequality. Since the distances to the east are written with positive numbers, the inequality does not need to be modified.
30<32d
These inequalities can be solved by using the Properties of Inequalities, a similar fashion to solving an equation. The first inequality is solved by multiplying by 2. 21d<-15
d<-30
30<32d
▼
Simplify
MultIneq
LHS⋅23<RHS⋅23
23⋅30<23⋅32d
MultFracByInverse
ba⋅ab=1
23⋅30<d
MoveRightFacToNum
ca⋅b=ca⋅b
23⋅30<d
Multiply
Multiply
290<d
CalcQuot
Calculate quotient
45<d
RearrangeIneq
Rearrange inequality
d>45
or.
d<-30 or d>45
b To graph the solution set of a compound inequality, it is convenient to first graph each individual inequality. First, the solution set of d<-30 is made of all the numbers to the left of -30, not including -30 because the inequality is strict.
Similarly, the graph of the solution set of d>45 is made of every number to the right of 45, not including 45.
Finally, since it is written with or,
the graph of the compound inequality is the combination of both solution sets and does not need to be adjusted or limited.
Example
Eat or Save Cookies?
Tearrik is spending his week of work experience at a local bakery. On his first day, he bought 100 cookies at a discount. He decides to eat 5 cookies a day until they are all gone. Tearrik is also allowed to bring home 3 cookies per day, which he gives to his brother. Tearrik's brother decides that he will not eat his cookies until he has at least 90 saved up.
a Let d be the number of days that have passed since Tearrik bought his cookies. Write a compound inequality to find on which days both brothers will eat cookies together.
b Graph the solution set of the inequality set in Part A.
Answer
a d<20 and d≥30
b 
Hint
a Rewrite the given information as individual inequalities.
b Graph the individual inequalities, then combine the graphs.
Solution
a To find on which days both brothers will eat cookies together, it is important to identify when each brother will eat cookies. Tearrik starts with 100 cookies and he eats 5 cookies every day until he reaches 0 cookies. This situation can be written as an inequality.
100−5d>0
Since Tearrik will eat cookies throughout this time, the values of d are the possible numbers of days in which he eats cookies. On the other hand, Tearrik's brother will save 3 cookies every day until he has at least 90 cookies. Then, Tearrik's brother will start eating his cookies. 3d≥90
For both to eat cookies, the value d must satisfy both inequalities at the same time. This kind of compound inequality is written with and.
100−5d>0 and 3d≥90
To solve this compound inequality, both inequalities must be solved. First, to solve the inequality on the left, 100 will be subtracted from both sides of the inequality.
100−5d>0
▼
Solve for d
d<20
d<20 and d≥30
This means that Tearrik will be able to eat cookies for 20 days, while his brother will save his cookies for 30 days before eating any of them. Therefore, they will not eat any cookies on the same day.
b To graph the solution set of the compound inequality, first it is necessary to graph the solution set of each individual inequality. The solution set of the inequality d<20 are the numbers less than 20, not including 20. Since 20 is not included in the solution set, it is marked with an open circle.
Similarly, the solution set of the inequality d≥30 is made of the numbers greater than or equal to 30. Since thi inequality is not strict, the circle is closed.
The solution set of the compound inequality is made of the numbers that satisfy both inequalities at the same time. Since there are no such numbers, the compound inequality has no solution.
This confirms that the brothers will not eat cookies together.
Example
Different Thermometers for Different Temperatures
For his internship, Davontay is working in a research lab. He is researching what tools are used to measure really low and really high temperatures. He found out that a thermocouple thermometer can measure temperatures lower than 3272∘ F, while a pyrometer thermometer can measure temperatures greater than or equal to 973 K.
To help find these temperatures in degrees Celsius, the relationships between the different temperature scales are shown in the following table. It should be noted that C refers to a temperature in degrees Celsius, F is the temperature in degrees Fahrenheit, and K refers to kelvins.
| Fahrenheit | Kelvin |
|---|---|
| 59C+32=F | C+273=K |
a What range of temperatures, in degrees Celsius, can be covered by using either thermometer?
b Graph the temperature range from Part A in a number line.
Answer
a C<1800 or C≥700
b 
Hint
a Rewrite the given information as inequalities.
b Graph the individual inequalities, then combine the graphs.
Solution
a To find the range of temperatures in degrees Celsius that can be covered by the thermometers, first it is needed to write the given information as inequalities. It is given that a thermocouple can measure temperatures that are lower than 3272∘ F. This can be written as an inequality, with F as the temperature in degrees Fahrenheit.
F<3272
Since the temperature is needed in degrees Celsius, the conversion formula will be substituted for F in the inequality above. Let C be the temperature in degrees Celsius.
59C+32<3272
On the other hand, it is given that a pyrometer can measure temperatures as low as 973 K. This can be written as an inequality, using K as the temperature in kelvins. It should be noted that kelvins are not degrees since it is an absolute scale.
K≥973
Substituting the conversion formula from the given table, the inequality can be rewritten in terms of degrees Celsius C. C+273≥973
Since the question asks to describe the temperature range that both thermometers can measure, if either one can be used, the compound inequality is written with or.
59C+32<3272 or C+273≥973
Now, to solve this compound inequality, each inequality will be solved individually using the Properties of Inequalities. To solve the inequality on the left, first subtract 32 from both sides of the inequality.
59C+32<3272
▼
Solve for C
SubIneq
LHS−32<RHS−32
59C<3240
MultIneq
LHS⋅95<RHS⋅95
C<95⋅3240
MoveRightFacToNum
ca⋅b=ca⋅b
C<95⋅3240
Multiply
Multiply
C<916200
CalcQuot
Calculate quotient
C<1800
C<1800 or C≥700
b To graph the temperature range from Part A, first the solution set of each individual inequality should be graphed. The solution set of the inequality C<1800 is made of all the numbers to the left of 1800, not including 1800.
Similarly, the solution set of the inequality C≥700 is made of every point to the right of 700, including 700.
Since the inequality is written with or,
the solution set of the compound inequality is made of the numbers that satisfy either inequality. By combining the graphs it can be noted that every number is a solution to the compound inequality.
This means that any temperature can be measured using either of the thermometers.
Example
Saving for a Car
Diego is helping at a local car dealership for his work experience. After spending time at the dealership, he decides to start saving money to buy a car. His father told him that he would double the amount of money that Diego saves, starting from now. Also, Diego will receive extra 500 dollars from his uncle to help buy the car when he finishes saving.
When looking for prices, Diego notices that most of the cars he likes range from 15 thousand dollars to 18 thousand dollars.
a How could the situation be written as a compound inequality?
b How much does Diego need to save to afford the car?
c Graph the inequality set in Part B.
Answer
a 15000≤2m+500≤18000
b Diego must save between $7250 and $8750, including these values, to afford the car.
c
Hint
a First write the total amount of money that Diego saves as an expression.
b Solve the compound inequality set in Part A.
c Graph each individual inequality, then combine the graphs.
Solution
a To write the situation as a compound inequality, first the total amount of money Diego needs to save should be expressed. Let m be the total amount of money that Diego saves. It is given that Diego's dad is going to double the amount of m. This means that m is multiplied by 2.
2m
Also, Diego's uncle will give Diego $500 dollars after finishing saving. This adds 500 to the amount after it is multiplied by 2. With this information, it is possible to write Diego's total amount of money as an expression.
2m+500
Diego needs from 15 thousand dollars to 18 thousand dollars to afford a car he likes. Therefore, the total amount of money that Diego needs must lie between these values in order for him to be able to afford the car. This can be expressed as a compound inequality.
2m+500≥15000 and 2m+500≤18000
Since the inequalities are written with and,they can be rewritten as follows.
15000≤2m+500≤18000
b To find the amount of money that Diego himself needs to save to be able to afford the car, the compound inequality set in Part A needs to be solved for m. This can be done using the Properties of Inequalities to solving each individual inequality first.
7250≤m≤8750
Since the inequality is non-strict, Diego can start looking to buy the car after he saves some amount of money from $7250 to $8750, including these values. After he saves enough, he can stop saving money.
c To graph the solution set of the compound inequality set in Part B, the solution set of each individual inequality will be graphed first. The solution set of m≥7250 is made of all the numbers to the right of 7250, including 7250.
Similarly, the graph of the solution set of inequality m≤8750 is made of all the points to the left of 8750, including 8750.
Finally, the solution set of the compound inequality is made of every number that both graphs share. In this case, the overlapping space from 7250 to 8750, inclusive.
Pop Quiz
Practice Solving Compound Inequalities
Various different compound inequalities will be shown in the applet below. Select the correct solution set.
Closure
Distances From Home to the Beach
At the beginning of the lesson, it was asked how far Vincenzo's house was from the beach. The following information was given.
- Vincenzo will drive at 50 miles per hour.
- He will arrive home in 40 to 55 minutes.
Then, the following questions were asked.
a How far can Vincenzo's house be from the beach? Round to one decimal place.
b Supposing Vincenzo's assumption is correct, let y be an invalid distance from his house to the beach. How could the possible values of y be written?
Answer
a Between 33.3 and 45.8 miles away.
b y<33.3 or y>45.8
Hint
a Write the situation as a compound inequality.
b Find the values that do not satisfy the compound inequality set in Part A.
Solution
a To find the distance from the beach to Vincenzo's house, a modification of the speed formula will be used.
r=td
In this formula, r is the rate or speed, d is the distance traveled, and t is the time duration of the movement. Since two time estimations are given, the time has to be the isolated quantity in the formula.
It is given that it will take Vincenzo between 40 and 55 minutes to get home. Therefore, the time t in the formula has to be at least 40. This is the same as writing that t is greater than or equal to 40 minutes. t≥40 minutes
However, since the speed is given in miles per hour, the 40 minute interval must be written in terms of hours. This can be done using a conversion factor.
40 minutes⋅60 minutes1 hour=32 hours
The inequality can be rewritten using this information.
t≥32 hours
Now the formula for the time in terms of the distance and the speed can be substituted into the inequality. It is given that Vincenzo will drive at 50 miles per hour. This means that r is equal to 50.
50d≥32 hours
The same can be done with the upper limit of 55 minutes. The first to note is that the statement can be written as t is less than or equal to 55 minutes
t≤55 minutes
Using the same conversion factor, the time can be rewritten in terms of hours.
55 minutes⋅60 minutes1 hour=1211 hours
And using this time span, the inequality can be rewritten.
t≤1211 hours
Finally, the expression in terms of the distance can be substituted into the inequality.
50d≤1211 hours
Since the possible values for d satisfy both inequalities, the compound inequality is written with and.
50d≥32 and 50d≤1211
Recall that compound inequalities written with andcan be rewritten as follows.
32≤50d≤1211
Finally, to find the possible values of d, each section of the compound inequality is multiplied by 50.
This means that Vincenzo's house is somewhere between 33.3 and 45.8 miles away from the beach. To graph the solution set of this compound inequality, each point between 33.3 and 45.8 will be included. Note that the compound inequality is non-strict, so the endpoints are denoted with closed circles. 
b From Part A, it was written that the possible values for d can be written using a compound inequality.
33.3≤d≤45.8
This compound inequality can be written with the word and.
d≥33.3 and d≤45.8
The values of y are the values that fall outside of this compound inequality, so each individual inequality should be reversed. Note that because 33.3 and 45.8 are included in the original interval, these values cannot be included in the solution set for y. Since d is greater than or equal to 33.3 miles, then y must be less than 33.3 miles.
y<33.3
Similarly, since the values of d are less than or equal to 45.8 miles, y has to be greater than 45.8 miles.
y>45.8
The possible values of y are those that are either less than 33.3 or greater than 45.8. Therefore, the compound inequality is written with or.
y<33.3 or y>45.8
The solution set of this compound inequality is made of every number left of 33.3 and right of 45.8. 
Solving and Graphing One Variable Compound Inequalities
Exercises
A snowboard shop sells snowboards whose lengths range from 139 cm to 180 cm. The shop has a sign that indicates that a snowboard should be about 0.88 times the height of the person, in centimeters.
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style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">3<\/span><span class=\"mord\">9<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">8<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">8<\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">8<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.78041em;vertical-align:-0.13597em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">3<\/span><span class=\"mord\">9<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">8<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">8<\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">8<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">3<\/span><span class=\"mord\">9<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">8<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">8<\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":3}
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">5<\/span><span class=\"mord\">8<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">0<\/span><span class=\"mord\">4<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.78041em;vertical-align:-0.13597em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">5<\/span><span class=\"mord\">8<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">0<\/span><span class=\"mord\">4<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7719400000000001em;vertical-align:-0.13597em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">5<\/span><span class=\"mord\">8<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">0<\/span><span class=\"mord\">4<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7719400000000001em;vertical-align:-0.13597em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.78041em;vertical-align:-0.13597em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">5<\/span><span class=\"mord\">8<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">0<\/span><span class=\"mord\">4<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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It is a given that the store sells snowboards which range in length from 139 centimeters to 180 centimeters. It is also said that the length of the snowboard should be 0.88 times the height of the person. If we let x represent the height of a person, we can write an expression for the recommended length of their snowboard. 0.88x In this shop, we cannot find a snowboard shorter than 139 centimeters or longer than 180 centimeters. We can represent the lengths of the snowboards not available in this shop with a compound inequality. 0.88x< 139 or 0.88x > 180
We are asked to solve the compound inequality set in Part A.
0.88x< 139 or 0.88x > 180
To solve a compound inequality, we need to solve each individual inequality separately. Let's do it beginning with the inequality at the left.
People with heights less than 158 centimeters cannot find a snowboard their size at this shop.
People taller than 204.5 centimeters cannot find a snowboard that suits them in this shop, according to the size chart. Now we can rewrite the compound inequality! x < 158 or x > 204.5
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Write a compound inequality for the graph.
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.7719400000000001em;vertical-align:-0.13597em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.78041em;vertical-align:-0.13597em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.78041em;vertical-align:-0.13597em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7719400000000001em;vertical-align:-0.13597em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's first look at where the shaded portion of the graph is. When a graph is shaded between two points, it represents an and
compound inequality. This is because the value of the variable must be greater than (or greater than or equal to) the lesser point and less than (or less than or equal to) the greater point.
Let the variable this compound inequality represents x and consider what inequalities could describe its value.
Lesser Point
The graph is shaded to the right of - 2, and the circle is open, so we can say that the value of x is greater than - 2. x> - 2
Greater Point
The graph is also shaded to the left of 4 and the circle is closed. This tells us that x is less than or equal to 4. x≤ 4
Compound Inequality
Notice that the solution set is sandwiched
between the two points. This tells us that we have an and
compound inequality. Rearranging x> - 2 will allow us to visualize this sandwich
when we write the compound inequality algebraically.
x> - 2 ⇔ - 2< x
Combining these two individual inequalities gives us a compound inequality: - 2 is less than x and x is less than or equal to 4.
- 2 < x and x ≤ 4
⇔ - 2< x ≤ 4
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Write a compound inequality for the graph.
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7719400000000001em;vertical-align:-0.13597em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.78041em;vertical-align:-0.13597em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">9<\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">9<\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">or<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7719400000000001em;vertical-align:-0.13597em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.88597em;vertical-align:-0.13597em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">9<\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">and<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.7891em;vertical-align:-0.0391em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">9<\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">and<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's first look at where the shaded portion of the graph is. When a number line is shaded with an un-shaded section between two points, it represents an or
compound inequality. This is because the value of the variable must be greater than (or greater than or equal to) the greater point or less than (or less than or equal to) the lesser point.
Let's call the variable this compound inequality represents x and consider what inequalities could describe its value.
Lesser Point
The graph is shaded to the left of -9 and the circle is closed. This portion tells us that x is less than or equal to -9. x≤ -9
Greater Point
The graph is also shaded to the right of - 5 and the circle is closed. This portion tells us that x is greater than or equal to - 5. x≥ - 5
Compound Inequality
Notice that the graph of this compound inequality is split
by an un-shaded region. This tells us that we have an or
compound inequality. The solution set has no region of overlap, so either x is less than or equal to -9 or x is greater than or equal to - 5.
x≤ -9 or x≥ -5
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Consider the following compound inequality.
4<3x+7≤13
Which is the correct graph of the solution set?
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We were asked to solve a compound inequality. Let's start by splitting it into separate inequalities.
Compound Inequality: 4 < 3&x + 7 ≤ 13
First Inequality: 4 < 3&x + 7
Second Inequality: 3&x + 7 ≤ 13
Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and.
4 < 3x + 7 and 3x + 7 ≤ 13
Let's solve the inequalities separately.
First Inequality
Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, we must flip the inequality sign.
This above tells us that - 1 is less than all values that satisfy the inequality.
Note that the point on 1 is open because it is not included in the solution set.
Second Inequality
Once more, we will solve the inequality by isolating the variable.
We found that all values less than or equal to 2 will satisfy the inequality.
Note that the point on 6 is closed because it is included in the solution set.
Compound Inequality
The solution set of the compound inequality is the intersection of the solution sets. First Solution Set: -1< x& Second Solution Set: x&≤ 2 Intersecting Solution Set: -1< x& ≤ 2 Finally, we will graph the solution set to the compound inequality on a number line.
Therefore, the answer is A.
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Which of the following inequalities are compound inequalities?
A.B.C.D.x>8, or x<-2x≤73<x≤246x>36 or -2x≥8
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We have been given four inequalities and are asked to decide whether they are compound. In order to decide, we need to understand what a compound inequality is. A compound inequality consists of two distinct inequalities joined by the word and or the word or. Now, we can start!
| Choice | Inequality | Explanation | Conclusion |
|---|---|---|---|
| A | x>8, or x<-2 | Two distinct inequalities joined by the word or. | Compound Inequality |
| B | x≤ 7 | This is a single inequality. | Not a Compound Inequality |
| C | 3< x≤ 24 ⇕ 3< x and x≤ 24 | Can be written as two distinct inequalities joined by the word and. | Compound Inequality |
| D | 6x>36 or -2x≥8 | Two distinct inequalities joined by the word or. | Compound Inequality |
The inequalities in A, C and D are compound.
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Solving and Graphing One Variable Compound Inequalities
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