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### Course: College Algebra>Unit 1

Lesson 4: Compound inequalities

# Compound inequalities: OR

Sal solves the compound inequality 5z+7<27 OR -3z≤18. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• why does he change the sign ? on the -3z< 18
• Because he divides by a negative number
• What is "P.E.M.D.A.S."?
• PEMDAS is the North American version of what we Europeans and Asians call BODMAS. All my peers here use PEMDAS, but I personally find that very confusing. It doesn't really matter what you use, you get the same answer.
Here are the full forms of the words
P- Parenthesis
E-exponents
M-multiplication
D-division
S-subtraction

While BODMAS is
B-Bracket
O-of
D-division
M-multiplication
S-subtraction

Hope it helped.
And uh... sorry for sending this 7 years late.
• I've taken the compound inequality practice a couple times now and haven't done well every time. I understand how to solve the problem, get each inequality's answer and plot it on a number line. What I don't understand is when to answer "No solutions", "All values of x are solutions", or the various answer choices based on the problem.
• Here's the scoop...
The word "OR" tells you to find the union of the 2 solution sets. The union is all the possible solutions from either inequality. So basically, a solution satisfies 1 or both of the individual inequalities.

There are 3 possible scenarios.
1) Solution is All real numbers. This is demonstrated in this video. You can see that the graph of the 2 inequalities ends up covering the entire number line.

2) The solution is 2 split intervals. For example: x<-2 OR x>0. The solution set is all numbers to the right of -2 combined with all the numbers larger than 0.

3) The solution is 1 interval. For example: x>-2 OR x>0. The union becomes x>-2 because this includes off the values from both inequalities.

If the inequality uses the word AND, then you need to find the intersection of the 2 solution sets. This is the values where the solution sets overlap (the values in common). Again, there are 3 scenarios.

1) Solution is No Solution. For example: x>5 AND x<0. These share no common values. When graphed, they have no overlap.

2) The 2 inequalities graph in opposite directions.
The solution is just where they overlap. For example: x>-2 AND x<0. The solution set is all numbers to the right of -2 up to the number 0. Basically, it is -2<x<0.

3) The 2 inequalities have graphs that go in the same direction. The solution becomes the shorter graph beause this is where they overlap. For example: x>-2 AND x>0. The intersection becomes x>0 because this includes the overlap (values in common) of both inequalities.

Hope this helps.
FYI - There is a video on union and intersection of sets. Use the search bar to find it. It may help you to understand the difference being OR (union) vs. AND (intersection).
• What exactly is the difference between AND/OR?
• AND tells you to find the intersection of the 2 solution sets. This is where the the solutions sets overlap, or what values they have in common.

OR tells you to find the union of the 2 solution sets. This is a new set that combines all values from both the solution sets.

Hope this helps.
• The test problems for these make no sense. On some of them the solution doesn't include numbers that both sets occupy on the number line, and for other ones they do like fpr x>5 or x <8 the solution will be all real numbers even though the numbers in between occupy the set, yet other times e.g. if, say, x>2 and x>4, then x>2 will be the solution even though someof the solutions will only be correct if the the number applies to both sets. What am I in missing?
• To understand this we need to look at the mathematical definitions of "and" and "or".

When we use "AND", then each statement must be true for all x.
Let's look at x>2 AND x>4.
If x< 2, say x=0 for example, then neither statement is true since 0>2 is not true and neither is 0>4, so x=0 CANNOT be in the solution set.
Lets say x=3 or x=4. Then 3>2 and 4>2 are TRUE but 3>4 and 4>4 are NOT, so neither 3 nor 4 are in the solution set. BUT, for any x>4, say 5, 5>2 is true and 5>4 is true, so 5 IS in the solution. So the answer in this case is all x such that x>4.

In the case of "OR", that means that AT LEAST ONE of the statements must be true.
Take x>5 or x<8. If x=0 then x<8 is satisfied, so even though x>5 is not satisfied, the statement x>5 or x<8 is true if x=0. Now, if x=7, then x>5 is true and x<8 is true, so the statement x>5 or x<8 when x=7 is also true and in this case, both conditions are satisfied (even though only one is required to be to make the OR true). Now, if x=10, then x>5 is true, x<8 is false, but since one of them is true, then x>5 or x<8 for x=10 is true.

Lets suppose that instead of x>5 or x<8, we had x<5 or x>8. In this case the statement would be false for all x when x=5,6,7,8. since for these values, neither x<5 or x>8 is true. But for any other x such that x is not a member of 5,6,7,8, the statement x<5 or x>8 is true.
Great Question!
Hope that helped.
Keep Studying.
• Quick question.
Is a 'OR' and 'AND' work the same way as it's in programming. Where OR mean if either one is true, and AND means both has to be true ?
• Yes, OR is when either have to be true and AND is when both have to be true.
• Why do we need compound inequalities? Do we use it in the real world?
• Compound inequalities are actually very important in life. Statisticians, business workers, engineers, and typical house-owners all use compound inequalities. They're kinda like systems of equations, which we use all the time in real life. But sometimes, our equations can't always equal something; they have to have constraints, like the value is greater than or less than, or something like that. Believe me, you will meet plenty of inequalities that can help you in life!
• Can someone summarize what Sal is trying to say here? I don't completely understand why both z<4 and z≥-6 will satisfy both inequalities. Help would be greatly appreciated! Thanks in advance! :)
• So for z<4 and z≥-6, Sal is basically saying that z is less than four, but greater than or equal to -6. Some possible solutions could be -5, -4, -3, -2, -1, 0, 1, 2, or 3.
• Can someone please explain what he just said?? I don't understand what the answer would be. I also don't understand why he graphed it and did all these really confusing steps. Sal NEVER explains things so that I can understand them. He's a good teacher, but I never understand.
• From what I got, he explained what x could be in a situation with 2 inequalities related by 'OR'. All of the numbers that can satisfy either the first inequality, the other, or both can be x. He was graphing a number line to show what each inequality's x could be so we could see an answer.
So first, he simplified the inequalities to make them easier to understand. Then, he showed what possibilities of x fulfilled the first inequality by the number line and did the same for the 2nd one too. With that in mind, he. showed that the whole number would satisfy one or both of the inequalities.
In conclusion, if there are 2 inequalities connected by 'OR', x can satisfy either or both of them. Simplifying the inequalities and graphing them on a number line shows the possible values of x.
• So, what would the answer be if both of the symbols were facing the same way?
• It would be the longer of the two arrows. For example: x>3 OR x>-5, the solution would be x>-5.

Your solution needs to include all the values from both original inequalities, which is why the longer arrow would be selected.

Hope this helps.

## Video transcript

Solve for z. 5z plus 7 is less than 27 or negative 3z is less than or equal to 18. So this is a compound inequality. We have two conditions here. So z can satisfy this or z can satisfy this over here. So let's just solve each of these inequalities. And just know that z can satisfy either of them. So let's just look at this. So if we look at just this one over here, we have 5z plus 7 is less than 27. Let's isolate the z's on the left-hand side. So let's subtract 7 from both sides to get rid of this 7 on the left-hand side. And so our left-hand side is just going to be 5z. Plus 7, minus 7-- those cancel out. 5z is less than 27 minus 7, is 20. So we have 5z is less than 20. Now we can divide both sides of this inequality by 5. And we don't have to swap the inequality because we're dividing by a positive number. And so we get z is less than 20/5. z is less than 4. Now, this was only one of the conditions. Let's [? look at ?] the other one over here. We have negative 3z is less than or equal to 18. Now, to isolate the z, we could just divide both sides of this inequality by negative 3. But remember, when you divide or multiply both sides of an inequality by a negative number, you have to swap the inequality. So we could write negative 3z. We're going to divide it by negative 3. And then you have 18. We're going to divide it by negative 3. But we're going to swap the inequality. So the less than or equal will become greater than or equal to. And so these guys cancel out. Negative 3 divided by negative 3 is 1. So we have z is greater than or equal to 18 over negative 3 is negative 6. And remember, it's this constraint or this constraint. And this constraint right over here boils down to this. And this one boils down to this. So our solution set-- z is less than 4 or z is greater than or equal to negative 6. So let me make this clear. Let me rewrite it. So z could be less than 4 or z is greater than or equal to negative 6. It can satisfy either one of these. And this is kind of interesting here. Let's plot these. So there's a number line right over there. Let's say that 0 is over here. We have 1, 2, 3, 4 is right over there. And then negative 6. We have 1, 2, 3, 4, 5, 6. That's negative 6 over there. Now, let's think about z being less than 4. We would put a circle around 4, since we're not including 4. And it'd be everything less than 4. Now let's think about what z being greater than or equal to negative 6 would mean. That means you can include negative 6. And it's everything-- let me do that in different color. It means you can include negative 6. I want to do that-- oh, here we go. It means you include negative 6. Let me do it in a more different color. Do it in orange. So z is greater than equal to negative 6. Means you can include negative 6. And it's everything greater than that, including 4. So it's everything greater than that. So what we see is we've essentially shaded in the entire number line. Every number will meet either one of these constraints or both of them. If we're over here, we're going to meet both of the constraints. If we're a number out here, we're going to meet this constraint. If we're a number down here, we're going to meet this constraint. And you could just try it out with a bunch of numbers. 0 will work. 0 plus 7 is 7, which is less than 27. And 3 times 0 is less than 18, so it meets both constraints. If we put 4 here, it should only meet one of the constraints. Negative 3 times 4 is negative 12, which is less than 18. So it meets this constraint, but it won't meet this constraint. Because you do 5 times 4 plus 7 is 27, which is not less than 27. It's equal to 27. Remember, this is an or. So you just have to meet one of the constraints. So 4 meets this constraint. So even 4 works. So it's really the entire number line will satisfy either one or both of these constraints.