If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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?

• • 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).
• 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.
• 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!
• 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 ? • 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! :) • I don't get the answer "all values of x"
The help section implies that it is everything on the graph, but if I go up to, let's say 10000, it can disprove one of the equations.
On a side note, when it is all values of x, does the value of x have to be the same for both of the inequalities? • • I am so confused on this. :( For this:

−15x+60≤105 AND 14x+11≤−31

why is the answer x = -3 ?

wouldn't it be "all values of x are solutions" ?

Every time I think I get this lesson I get one wrong. Why are inequalities so elusive and confusing? It seems like the rules are always changing! I am sad. Can someone help plz :(  