SOLUTION: in compound inequalities, say you were given a disjunction inequality that read: -y +5 ≥ 9 or 3y + 4 < -5. when solved, you’d get: y ≤ -4 or y < -3. why would you only writ

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Question 1172851: in compound inequalities, say you were given a disjunction inequality that read: -y +5 ≥ 9 or 3y + 4 < -5. when solved, you’d get: y ≤ -4 or y < -3. why would you only write one of the inequalities as your final answer, rather than one since it’s connected by, “or”? as with a conjunction inequality, if they gave you for example, k > 1 and k > 5 (both signs are facing the same direction), why would you only write one of the inequalities? if there’s a rule that corresponds with this, please tell me. what/when does the rule apply to (if any)? does this only happen in conjunction inequalities if both signs are facing the same direction and one of the inequalities can “fit” under the other? fit meaning that (referring back to the previous example) all numbers greater than 1 aren’t all greater than 0, but all numbers greater than 5 are greater than 1? please respond. i have a math test on this soon, thank you 😄
Found 5 solutions by math_tutor2020, Edwin McCravy, mccravyedwin, AnlytcPhil, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Using a number line may help visualize what's going on.

Graph y ≤ -4 by plotting a closed circle at -4 on the number line and then shade to the left. This indicates "y is -4 or smaller".

To graph y < -3, we will plot an open circle at -3 and shade to the left. The open circle says "do not include this value as part of the solution set".

Since we're combining those inequalities with the "or" keyword, this means we're looking for y values that are in either region.
So either we're in region A which has y = -4 or smaller OR we're in region B which describes y values smaller than -3.

We can simplify those two regions to combine them to simply region B.
As the diagram shows, if you are in region A (red), you're already in region B (blue), but not the other way around


---------------------------------------------------------

Now for the next example:
k > 1 and k > 5

To graph k > 1, we have an open hole at 1 and shading to the right to describe all values larger than 1. The value 1 is not part of the solution set.

The graph of k > 5 is similar, but the open hole is at 5 this time.

The keyword here is "and" this time. This means we need to be in BOTH regions at the same time. So we'll look where they overlap. The two regions overlap beyond 5. So that's why "k > 1 and k > 5" simplifies to "k > 5".

If you pick a number that's both larger than 1 AND larger than 5, then that number is larger than 5. This is because 5 is larger than 1.


Hopefully these visual diagrams clear your question up. If not, then please let me know.

Answer by Edwin McCravy(20066)   (Show Source): You can put this solution on YOUR website!
 -y +5 ≥ 9 or 3y + 4 < -5.
    -y ≥ 4 or     3y < -9. 
    y ≤ -4    or   y < -3

If you say y < -3 then you don't need to say y ≤ -4 because even if y is ≤ -4,
it's also < -3.  So y < -3 covers all the possible cases.

Edwin
Answer by mccravyedwin(409)   (Show Source): You can put this solution on YOUR website!
I should also point out that if the word "and" were between them instead of "or",

-y + 5 ≥ 9 and 3y + 4 < -5.
    -y ≥ 4 and     3y < -9. 
    y ≤ -4    and   y < -3

If you say y ≤ -4 then you don't need to say y < -3 because if y is ≤ -4,
it's automatically < -3.  So y ≤ -4 covers all the possible cases.

Edwin
Answer by AnlytcPhil(1810)   (Show Source): You can put this solution on YOUR website!

Answer by ikleyn(52932)   (Show Source): You can put this solution on YOUR website!
.

Shortly speaking, when you have two inequalities


    y <= -4   OR   y < -3


connected with the service word  " OR ",  it means that  the solution " y " belongs 

to the union of the sets   (-oo,-4]  and  (-oo,-3).


Of these two sets, one is embedded to another. THEREFORE, their union is  (-oo,-3),  the larger of the two sets.





In other case, when you have two inequalities


    k > 1   AND   k > 5


connected with the service word  " AND ",  it means that  the solution " k " belongs 

to the INTERSECTION of the sets   (1,oo)  and  (5,oo).


Of these two sets, one is embedded to another. THEREFORE, their intersection is  (5,oo),  the smaller of the two sets.




So, the difference between these cases  is in the SERVICE WORD  "OR"  or  "AND" used.

You should ALWAYS look ATTENTIVELY on what service word is used in your case,

and then take EITHER the union OR the intersection of separate sets, appropriately.


It is  THE  SAME  explanation as you get from other tutor,  but expressed in other words and in the shorter form.


---------------

See the lessons
    - Solving systems of linear inequalities in one unknown
    - Solving compound inequalities
in this site.

Do not miss other relevant lessons on inequalities - the links are at the end of the referred lessons.

. . . . . . .

Let us know about the results of your test.


Come again to this forum soon with your questions to learn something new (!)

Do not postpone your visit and your questions to the last day . . .



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