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### Course: Algebra 1>Unit 2

Lesson 6: Compound inequalities

# Compound inequalities: AND

Sal solves the compound inequality 3y+7<2y AND 4y+8>-48. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• If we got that y < -7; and y > -14, is it correct to state it like this: -14 < y < -7 ??
• Yes, that is the most common way of expressing that type of solution. You could also just leave it as an "and" statement as you did in the first half of your comment, or even put it in set and interval notation. They all mean the same thing, and it just comes down to the visual representation that you click with the best (or that your teacher or test requests).
• what is the difference between compound inequalities one and two
• The first equation he did in "Compound Inequalities 1 video" had a "or" in between the two inequalities while this video has a "and" in it.

"Or" means it can satisfy either one of the inequalities or both while "and" means it has to satisfy both inequalities for x to qualify. You can do this by inserting a number that falls in the number line that you made for x, or after finding "x" inserting one of the numbers that x could be.
• I am very confused. How do you answer the questions? I already know how to break down an inequality to x<7 from 3x+4>25 or something like that, but if it gives me a question like:
2x+3≥7 OR 2x+9>11
Where inequality 1 breaks down to x≥2 and in equality 2 breaks down to x>1, how would the answer be
x>1? I don't get it, because couldn't the answer very well be x≥2 because it says OR?? I am very confused. Please help me.
• This is a very good question! In a problem where it says OR, either of the equalities or both equalities can satisfy the equation. In this instance, x>1, when graphed on a number line, and since the equality is greater than 1, x>1 definitely satisfies both equalities because its line encompasses the other equality's line. I would encourage you to make a number line and graph the two equalities to visualize them. This may help to alleviate your confusion.
• what happens when you divide a negative by a negative and isolate it?
• Here's a chart to help you.
By the way, N means negative and P is positive.

N*P=N
P*N=N
N*N=P
P*P=P

ta da!
• Can someone please explain how he gets the answers? I know how to do the math just not pick the answers. Like if it's no solution, or all the values of x are solutions.
• The AND tells you to find the intersection of the two solution sets. Any value put in the intersection must be in both of the original solution sets. It's the values in common, or where the two sets overlap. An intersection is very unlikely to create a solution of all real numbers. The only way that would be possible is if each individual inequality also had solutions of all real numbers. You can get no solution with an intersection because the two solution sets may have no values in common.

An OR tells you to find the union of the two solution sets. This will combine all solutions from both sets. So, any solution that works in one or both sets is part of the union. A union can create a solution of all real numbers.

Hope this helps.
• I'm having a lot of trouble with the exercise after this. I don't have a problem with solving the inequalities themselves, but I don't understand the part when it asks if there's no answer, or if all values of x are answers, or if x is more than/less than y, etc. Can someone help me?
• where did -10 come from?
• It's just a random number he chose that was between -14 and -7.
• This is very helpful and all but , what if i'm dealing with inequalities that for example are like this,
-4x < 16
do i use the same exact operation? and if yes explain why and if not, explain how.
• You have a single inequality, not a compound inequality. you solve individual inequalities the same way you solve equations. You use opposite operations to move items across the inequality. In your inequality, you divide both sides by -4. The one thing you need to do differently from an equations is that when you divide / multiply by a negative, you reverse the inequality.

Compound inequalities consist of 2 or more inequalities joined with the word AND or OR. You don't have that situation.
• Wait, how are AND inequalities different from OR inequalities?
• Let me explain with an example. Let's take two inequalities
x<3, x>1
If the two inequalities are joined by AND, both of the inequalities must be satisfied by the values of x. In other words, both the inequalities must be true at the same time.
x<3 AND x>1 means x must be smaller than 3 and x must be larger than 1. Clearly x must lie between 1 and 3 so x∈(1,3).
If the two inequalities are joined by OR, the inequality will be true even if the value of x is true for one inequality and false for the other inequality.
x<3 OR x>1 means that x is less than 3 or x is greater than 1. Since any one of these possibilities is true for every real number, x∈R.
In essence, when using AND to join 2 inequalities we take the intersection of the solution sets of the 2 inequalities and when using OR to join 2 inequalities we take the union of the solution sets of the 2 inequalities.
• I don't understand where you got those numbers for the number line I am very confused do I need that or was it just to help in something?
• To solve a compound inequality, you start by solving each individual inequality. Then, the word "AND" or "OR" tells you the next step to take.

AND tells you to find the intersection of the two solution sets. An intersection is the values in common or the overlap of the two sets. This is why it is common to graph the 2 original inequalities. From the graph, you can quickly identify what values are in common because it is where the graphs overlap.

OR tells you to find the union of the two solution sets. A union combines all solutions from the original inequalities into one solution set. If a value works for either inequality or both inequalities, it goes in the union.

For more info on Intersections vs Unions, try this video: https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/intersection-and-union-of-sets

Hope this helps.