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.

### Course: Algebra 1>Unit 2

Lesson 6: 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.
• 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.
• 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!
• Can there be an OR inequality that has no solutions?
• The only way for an OR inequality to have no solution is for both individual inequalities to have no solution. Basically, they both are contradictions (false statements).
• how do you choose an answer to an 'or' compound inequality? Since you get two answers, how do you know which one is correct?
with lots of thanks,
Izzy.
• The word OR tells you to find the union of the 2 solutions sets from the individual inequalities. It combines both sets into one. As long as a solution works in at least one of the inequalities, then it is a solution to the compound inequality. You don't pick one set of answers over the other.
Hope this helps.
• 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 : )