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## College Algebra

### Course: College Algebra>Unit 1

Lesson 4: Compound inequalities

# A compound inequality with no solution

Sal solves the compound inequality 5x-3<12 AND 4x+1>25, only to realize there's no x-value that makes both inequalities true. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Hi! So my question is more so regarding the questions section that you usually do to test yourself after watching the videos. I’ve been trying to finish it with a perfect score for the past two days but I simply do not get the thinking behind the answer choices. I know how to solve the inequality, I know how to graph it, but when it asks me to pick the right answer between both solutions I become completely confused! Not to mention the other answer choices such as: solution for inequality A, solution for inequality B, solution for both, “All x’s are right”, or “no solution” the answer always surprises me and the hint section is not helping. Would someone explain to me how to get past it? Would it be possible for Sal to make a short video on how to solve the questions and pick between those answers?

Thank you and sorry for the lengthy post!
• Sounds like you are getting confused when you have to figure out the intersection or the union of the 2 inequalities. There is a video on intersections and unions of sets. This might help you understand the basic concept of intersections and unions. It is at this link: https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/intersection-and-union-of-sets

The easiest way I find to do the intersection or the union of the 2 inequalities is to graph both.
If the compound inequality is "and", you need to find the intersection. The intersection is where the values of the 2 inequalities overlap. For example:
-- graph x > -2 and x < 5. These overlap from -2 up to 5. The intersection is: -2 < x < 5; or in interval notation: (-2, 5)
-- graph x > -2 and x > 5. These 2 inequalities overlap for all values larger than 5. The intersection is: x>5; or in interval notation: (5, infinity)
-- graph x > -2 and x < -5. These 2 inequalities have no overlap. So, there is no intersection. This is the case that results in No Solution.

If the compound inequality is "or", you need to find the union. The union of the 2 inequalities is a new set that contains all values from both sets combined. For example:
-- graph x > -2 or x < -5. The 2 inequalities have completely separate graphs. All values from both graphs become the solution: x > -2 or x < -5; or in interval notation: (-infinity, -5) or (-2, infinity)
-- graph x > -2 or x > 5. The graphs of the inequalities go in the same direction. We need a set that includes all values for both inequalities. This would be the longer graph. So, the solution is: x > -2; or in interval notation: (-2, infinity)
-- graph x > -2 or x < 5. These overlap -- so the union of the 2 sets would encompass the entire number line. This is the scenario that become All Real Numbers or All values of X are solutions.

hope this helps.
• Is it possible to graph a no solution inequality on the number line? If so, how?
• No, it can't be graphed, since if there is no solution, there is nothing to put on the graph!
• Sal states that there is no solution, but what if x was a function of some sorts or a liner equation with multiple places on the number line that fall into the constraints both less then 3 and greater than 6? example, a solution set of (2,7)
• A set of values cannot satisfy different parts of an inequality of real numbers. The variable is a real number here. If you wanted to specify an inequality that described functions, you would have something very different.
• my question is whats the point of this. when will i use this in the real world lmao
• When buying groceries in the future, you might get asked this question. Hence, it's important to always know how to do it!
• At that point couldn't you bend the number line like you can bend space?
I know you can't, but still.
• how do you know when to switch the inequality symbol?
• You only switch the inequality symbol when you are multiplying or dividing by a negative. Hope this helps : )
• How do you eliminate options in the problems. What is the difference between AND and OR? I am REALLY struggling with this concept. Its like math block. I feel like I've never struggled more with a concept than this one. AAAH! Please help.
• The word AND tells you to find the intersection of both solution sets. An intersection is the solutions in common, or that overlab.

The word OR tells you to find the union of the 2 solution sets. A union is 2 sets combine all possible solutions from both sets.

To learn more about these, search for "intersection and union of sets". There is a video on KA that walks you thru them.
• if there is no solution then how come there was two findings for x.
• Each individual inequality has a solution set.
But the word "and" in the compound inequality tells us to find the intersection of those 2 solution sets. The intersection is the final solution for the whole problem.
An intersection of 2 sets is where the sets overlap (or which values are in common). If you graph the 2 inequality solutions, you can see that they have no values in common. There is no overlap in their 2 sets. This is why the compound inequality has no solution.

• @ sal says that there is no solution to the example equation, but i was wondering if it did have a solution like 1/ 0 as anything by zero gives infinity or negative infinity. really crazy question but just asking
• Let's assume that when solving for any equation - or "x" in this case - the answer comes out to be "1/0". If this happens, the answer is thus undefined and there is no solution.

Therefore, to help you clarify, anything divided by zero - as with the case of 1/0 - is NOT infinity or negative infinity. It is simply undefined. More accurately, it would be better to say in your above statement that anything which APPROACHES 1/0 is positive infinity or negative infinity. Numbers that approach 1/0 would be something like "1/0.1", "1/0.001", or "1/0.000001" - where the last example number would equal to 1,000,000. So, here in the example, we are able to show that as the denominator get closer and closer to zero, the fraction as a whole get closer and closer to a really BIG number - or infinity. However, when the denominator becomes zero, it is NOT infinity but an undefined number.

However, if you want to get more in-depth, here's an amazing and easy to follow animated TED-Ed video that explains the whole idea in less than five minutes REALLY well: https://www.youtube.com/watch?v=NKmGVE85GUU

Hope this helps!
• can there be a no solution for an OR compound inequality or is it just for AND compound inequalities?
**If YES to no solution for OR compound inequalities can you provide an example Please?
• If any of the inequalities in the compound OR inequality have a valid solution, the compound OR inequality will also have a valid solution.

The inequality below has no solutions because x^2 + 1 is never less than 0 and -x^2 - x - 2 is never greater than 0.
x^2 + 1 < 0 OR -x^2 - x - 2 > 0

## Video transcript

Solve for x, 5x - 3 is less than 12 "and" 4x plus 1 is greater than 25. So let's just solve for X in each of these constraints and keep in mind that any x has to satisfy both of them because it's an "and" over here so first we have this 5 x minus 3 is less than 12 so if we want to isolate the x we can get rid of this negative 3 here by adding 3 to both sides so let's add 3 to both sides of this inequality. The left-hand side, we're just left with a 5x, the minus 3 and the plus 3 cancel out. 5x is less than 12 plus 3 is 15. Now we can divide both sides by positive 5, that won't swap the inequality since 5 is positive. So we divide both sides by positive 5 and we are left with just from this constraint that x is less than 15 over 5, which is 3. So that constraint over here. But we have the second constraint as well. We have this one, we have 4x plus 1 is greater than 25. So very similarly we can subtract one from both sides to get rid of that one on the left-hand side. And we get 4x, the ones cancel out. is greater than 25 minus one is 24. Divide both sides by positive 4 Don't have to do anything to the inequality since it's a positive number. And we get x is greater than 24 over 4 is 6. And remember there was that "and" over here. We have this "and". So x has to be less than 3 "and" x has to be greater than 6. So already your brain might be realizing that this is a little bit strange. This first constraint says that x needs to be less than 3 so this is 3 on the number line. We're saying x has to be less than 3 so it has to be in this shaded area right over there. This second constraint says that x has to be greater than 6. So if this is 6 over here, it says that x has to greater than 6. It can't even include 6. And since we have this "and" here. The only x-es that are a solution for this compound inequality are the ones that satisfy both. The ones that are in the overlap of their solution set. But when you look at it right over here it's clear that there is no overlap. There is no x that is both greater than 6 "and" less than 3. So in this situation we have no solution.