I don't quite understand the union part of compound inequalities. Even if I find the union of an AND inequality, it seems like more often than not the union solution does not apply to both of the inequalities.

compound inequalities are algebra inequalities but, use a line plot. So a inequality first should be solved. Then you must make a line plot. Then you must plot the line plot and show that it is greater than, less than and something like that. Anyway after the line plot is done you must answer the question meaning what x or z or y or ect. means. I do not know what the *union* you are talking about. Please explain.* I hope this helps, please tell me if you have any more questions and please ask more questions. 😏😊😉😄😃😅🙄🙄🙄 Ask more questions.

how do you solve it with fracttions involved?

I'll take an example here. Let's say you have 3 <1/2x + 5 < 17. So you would solve this by first subtracting five from both sides using the Subtraction Property of Equality, which would look like this: 3 - 5 < 1/2x + 5 - 5 < 17 - 5 --> -2 < 1/2x < 12. From there, you want to change the 1/2x into x (or 1x), and to do that, you would multiply the fraction to get 1 in order to isolate the variable (for instance, 2/3 would be multiplied by 3/2 to get 1), so the end result would look like this: -2*2 < 1/2x*2 < 12*2, which would lead to the answer of -4 < x < 24.

Can someone give an example of AND that has "many solutions"? I'll be the one to graph it. I just need equations. Thank you.

It could also be as simple as -6 ≤ 2x + 2 ≤ 8 which says that 2x + 2 ≥ - 6 AND 2x + 2 ≤ 8. To solve, subtract 2 all the way across, -8 ≤ 2x ≤ 6, then divide by 2 to get -4 ≤ x ≤ 3. Infinite amount of numbers between -4 and 3.

How would I write an "or" compound inequality?

You just write OR between the inequality like 4>x OR x>2 It is not like and were it is 4>x>2

So in a OR inequality, the answer can not be “there are no solutions?

That's correct. Any solution from either individual inequality becomes a solution for the compound inequality. Similarly, with an AND inequality, you can't get a solution of all real numbers because the AND restricts the solution set to only those value in common to both individual inequalities.

Does the presence of a solution to an inequality depend on whether if the inequality is an 'OR' and an 'AND'?

Yes! The AND and OR are operations themselves. The AND tells you to find the intersection of the 2 solution sets (final solution is only the solutions in common to both inequalities). The OR tells you to find the union of the 2 solutions sets (all possible solutions to either inequality)

8v - 7 ≤ 10 - 9v ≤ -8v + 10

Break it into two parts and put back together later. So 8v-7≤10-9y and 10-9v≤-8v+10. On the first, add 9y and add 7 to get 17v ≤ 17, so v ≤1. On the second, you add 9v and subtract 10 to get 0≤v. Putting these together as a union, 0≤v≤1. Values in the range such as 0 gives -7≤10≤10 which is a true statement and 1 gives 1≤1≤2 which is also true. Trying -1 (-15≤19≤18) and 2 (9≤-8≤-6) both of which are false.

5x−19≤1 OR −4x+3<−6

Did you try to solve this? Your problem is very similar to the example in: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities/x2f8bb11595b61c86:compound-inequalities/v/compund-inequalities 1) Solve the individual inequalities 2) The "or" tells you to do the union of the 2 solution sets. A union includes any value that solves one or both of the inequalities. Graph both inequalities to find their union. Give the problem a try. Comment back if you have questions or want me to check your result.

Main content

Algebra 1

Course: Algebra 1 > Unit 2

Lesson 6: Compound inequalities

Compound inequalities review

Q: Can someone give an example of AND that has "many solutions"? I'll be the one to graph it. I just need equations. Thank you.

It could also be as simple as -6 ≤ 2x + 2 ≤ 8 which says that 2x + 2 ≥ - 6 AND 2x + 2 ≤ 8. To solve, subtract 2 all the way across, -8 ≤ 2x ≤ 6, then divide by 2 to get -4 ≤ x ≤ 3. Infinite amount of numbers between -4 and 3.

Q: How would I write an "or" compound inequality?

You just write OR between the inequality like 4>x OR x>2 It is not like and were it is 4>x>2

Q: 5x−19≤1 OR −4x+3<−6

Did you try to solve this? Your problem is very similar to the example in: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities/x2f8bb11595b61c86:compound-inequalities/v/compund-inequalities 1) Solve the individual inequalities 2) The "or" tells you to do the union of the 2 solution sets. A union includes any value that solves one or both of the inequalities. Graph both inequalities to find their union. Give the problem a try. Comment back if you have questions or want me to check your result.

CCSS.Math:

HSA.REI.B.3

Google Classroom

A compound inequality is an inequality that combines two simple inequalities. This article provides a review of how to graph and solve compound inequalities.

What is a compound inequality?

A compound inequality is an inequality that combines two simple inequalities. Let's take a look at some examples.

Example with "OR"

x, is less than, 3, space, start color #7854ab, start text, space, O, R, space, end text, end color #7854ab, space, x, is greater than, 5

So, for example, the numbers 0 and 6 are both solutions of the compound inequality, but the number 4 is not a solution.

Example with "AND"

x, is greater than, 0, space, start color #e07d10, start text, space, A, N, D, space, end text, end color #e07d10, space, x, is less than, 4

This compound inequality is true for values that are both greater than zero and less than four. Graphically, we represent it like this:

So, in this case, 2 is a solution of the compound inequality, but 5 is not because it only satisfies one of the inequalities, not both.

Note: If we wanted to, we could write this compound inequality more simply like this:

0, is less than, x, is less than, 4

Solving compound inequalities

Example with "OR"

Solve for x.

2, x, plus, 3, is greater than or equal to, 7, space, start color #7854ab, start text, space, O, R, space, end text, end color #7854ab, space, 2, x, plus, 9, is greater than, 11

Solving the first inequality for x, we get:

\begin{aligned} 2x+3 &\geq 7 \\\\ 2x &\geq 4 \\\\ x &\geq 2 \end{aligned}

Solving the second inequality for x, we get:

\begin{aligned} 2x+9&>11 \\\\ 2x&>2\\\\ x&>1 \end{aligned}

Graphically, we get:

So our compound inequality can be expressed as the simple inequality:

x, is greater than, 1

Want to learn more about compound inequalities that use OR statements? Check out this video.

Example with "AND"

Solve for x.

4, x, minus, 39, is greater than, minus, 43, space, start color #e07d10, start text, space, A, N, D, end text, end color #e07d10, space, 8, x, plus, 31, is less than, 23

Solving the first inequality for x, we get:

\begin{aligned}4x-39&> -43 \\\\ 4x &> -4 \\\\ x &>-1 \end{aligned}

Solving the second inequality for x, we get:

\begin{aligned} 8x+31&<23\\\\ 8x&<-8\\\\ x&<-1 \end{aligned}

Graphically, we get:

Strangely, this means that there are no solutions to the compound inequality because there's no value of x that's both greater than negative one and less than negative one.

Want to learn more about compound inequalities that use AND statements? Check out this video.

Practice

Problem 1

Current

Solve for x.

5, x, minus, 4, is greater than or equal to, 12, space, start text, space, O, R, space, end text, space, 12, x, plus, 5, is less than or equal to, minus, 4

(Choice A)
x, is greater than or equal to, start fraction, 16, divided by, 5, end fraction or x, is less than or equal to, minus, start fraction, 3, divided by, 4, end fraction
(Choice B)
x, is less than or equal to, start fraction, 16, divided by, 5, end fraction
(Choice C)
x, is greater than or equal to, minus, start fraction, 3, divided by, 4, end fraction
(Choice D)
There are no solutions
(Choice E)
All values of x are solutions

Want more practice? Check out this exercise.

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Miguel Monteon :/
Posted 6 years ago. Direct link to Miguel Monteon :/'s post “how do you solve it with ...”
how do you solve it with fracttions involved?
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(6 votes)
Answer
- Agent Raf
  Posted 6 years ago. Direct link to Agent Raf's post “I'll take an example here...”
  I'll take an example here. Let's say you have 3 <1/2x + 5 < 17. So you would solve this by first subtracting five from both sides using the Subtraction Property of Equality, which would look like this: 3 - 5 < 1/2x + 5 - 5 < 17 - 5 --> -2 < 1/2x < 12. From there, you want to change the 1/2x into x (or 1x), and to do that, you would multiply the fraction to get 1 in order to isolate the variable (for instance, 2/3 would be multiplied by 3/2 to get 1), so the end result would look like this: -2*2 < 1/2x*2 < 12*2, which would lead to the answer of -4 < x < 24.
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  (35 votes)
Bart Kornacki
Posted 6 years ago. Direct link to Bart Kornacki's post “Why didn't you swap an in...”
Why didn't you swap an inequality with a sign in problem 2 practice while dividing by negative number (-7). This has changed the equation!
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(10 votes)
Answer
nekk
Posted 4 years ago. Direct link to nekk's post “I don't quite understand ...”
I don't quite understand the union part of compound inequalities.
Even if I find the union of an AND inequality, it seems like more often than not the union solution does not apply to both of the inequalities.
Button navigates to signup pageButton navigates to signup page
(7 votes)
Answer
- STiger24
  Posted 4 years ago. Direct link to STiger24's post “compound inequalities are...”
  compound inequalities are algebra inequalities but, use a line plot.
  
  So a inequality first should be solved.
  
  Then you must make a line plot.
  
  Then you must plot the line plot and show that it is greater than, less than and something like that.
  
  Anyway after the line plot is done you must answer the question meaning what x or z or y or ect. means.
  
  I do not know what the union you are talking about.
  
  Please explain.*
  
  I hope this helps, please tell me if you have any more questions and please ask more questions.
  
  😏😊😉😄😃😅🙄🙄🙄
  
  Ask more questions.
  Comment on STiger24's post “compound inequalities are...”
  (11 votes)
yoonjuliagil
Posted 3 years ago. Direct link to yoonjuliagil's post “Does the presence of a so...”
Does the presence of a solution to an inequality depend on whether if the inequality is an 'OR' and an 'AND'?
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(4 votes)
Answer
- Kim Seidel
  Posted 3 years ago. Direct link to Kim Seidel's post “Yes! The AND and OR are ...”
  Yes! The AND and OR are operations themselves. The AND tells you to find the intersection of the 2 solution sets (final solution is only the solutions in common to both inequalities). The OR tells you to find the union of the 2 solutions sets (all possible solutions to either inequality)
  Button navigates to signup page
  (14 votes)
Josh
Posted 5 years ago. Direct link to Josh's post “Can someone give an examp...”
Can someone give an example of AND that has "many solutions"? I'll be the one to graph it. I just need equations. Thank you.
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- David Severin
  Posted 5 years ago. Direct link to David Severin's post “It could also be as simpl...”
  It could also be as simple as -6 ≤ 2x + 2 ≤ 8 which says that 2x + 2 ≥ - 6 AND 2x + 2 ≤ 8. To solve, subtract 2 all the way across, -8 ≤ 2x ≤ 6, then divide by 2 to get -4 ≤ x ≤ 3. Infinite amount of numbers between -4 and 3.
  Comment on David Severin's post “It could also be as simpl...”
  (2 votes)
Owen Choi
Posted 10 days ago. Direct link to Owen Choi's post “Instead of focusing on wh...”
Instead of focusing on why math is bad, let's explore how we can improve our understanding and appreciation of this important subject. Math is an essential part of our daily lives, and it helps us understand the world around us. It is a valuable tool for problem-solving, critical thinking, and decision-making. Is there anything specific about math that you find challenging or confusing? I'm here to help and provide guidance.
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(9 votes)
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Christy Brokaw
Posted 5 years ago. Direct link to Christy Brokaw's post “How would I write an "or"...”
How would I write an "or" compound inequality?
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(5 votes)
Answer
- Ben
  Posted 5 years ago. Direct link to Ben's post “You just write OR between...”
  You just write OR between the inequality like
  
  4>x OR x>2 It is not like and were it is 4>x>2
  Button navigates to signup page
  (4 votes)
Ethan hernandez
Posted 4 years ago. Direct link to Ethan hernandez's post “8v - 7 ≤ 10 - 9v ≤ -8v + ...”
8v - 7 ≤ 10 - 9v ≤ -8v + 10
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(4 votes)
Answer
- David Severin
  Posted 4 years ago. Direct link to David Severin's post “Break it into two parts a...”
  Break it into two parts and put back together later. So 8v-7≤10-9y and 10-9v≤-8v+10. On the first, add 9y and add 7 to get 17v ≤ 17, so v ≤1. On the second, you add 9v and subtract 10 to get 0≤v. Putting these together as a union, 0≤v≤1. Values in the range such as 0 gives -7≤10≤10 which is a true statement and 1 gives 1≤1≤2 which is also true. Trying -1 (-15≤19≤18) and 2 (9≤-8≤-6) both of which are false.
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  (6 votes)
Proplayer
Posted a month ago. Direct link to Proplayer's post “So in a OR inequality, th...”
So in a OR inequality, the answer can not be “there are no solutions?
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(5 votes)
Answer
- Kim Seidel
  Posted a month ago. Direct link to Kim Seidel's post “That's correct. Any solu...”
  That's correct. Any solution from either individual inequality becomes a solution for the compound inequality.
  
  Similarly, with an AND inequality, you can't get a solution of all real numbers because the AND restricts the solution set to only those value in common to both individual inequalities.
  Button navigates to signup page
  (4 votes)
GABRIELLES
Posted 2 years ago. Direct link to GABRIELLES's post “5x−19≤1 OR −4x+3<−6”
5x−19≤1 OR −4x+3<−6
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(4 votes)
Answer
- Kim Seidel
  Posted 2 years ago. Direct link to Kim Seidel's post “Did you try to solve this...”
  Did you try to solve this? Your problem is very similar to the example in: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities/x2f8bb11595b61c86:compound-inequalities/v/compund-inequalities
  
  1) Solve the individual inequalities
  2) The "or" tells you to do the union of the 2 solution sets. A union includes any value that solves one or both of the inequalities. Graph both inequalities to find their union.
  
  Give the problem a try. Comment back if you have questions or want me to check your result.
  Button navigates to signup page
  (4 votes)