AS Maths Inequalities

Q: Use set notation to describe the set of values of x for which:

x^2 - x - 6 > 0 and 10 - 2x < 5
———

I got the values x < -2 , x > 3 , and x > 2.5

The markscheme says the answer is x > 3
But I don’t understand why. Any help would be greatly appreciated 🙏

Reply 1

tonyiptony

Be careful with the connector "and" (in contrast to "or"). "And" means you want both inequalities to hold at the same time.
A sketch of the solutions on the real line should tell you when are both true.
Alternatively, depending on how good you are at set notation manipulations, it comes down to fairly routine computation. (EDIT: Definitely don't need it here, IMO. A nice picture should do the job.)

As a sanity check, you say x<-2 works. Well, x=-3 doesn't fit the second inequality, so something must have gone wrong.

Moral of the story: The words "and" and "or" in mathematics means something quite specific. In fact, a lot of words have a specific meaning in maths.

(edited 6 months ago)

Reply 2

rhyso4

Original post by tonyiptony

Be careful with the connector "and" (in contrast to "or"). "And" means you want both inequalities to hold at the same time.
A sketch of the solutions on the real line should tell you when are both true.
Alternatively, depending on how good you are at set notation manipulations, it comes down to fairly routine computation.

As a sanity check, you say x<-2 works. Well, x=-3 doesn't fit the second inequality, so something must have gone wrong.

Moral of the story: The words "and" and "or" in mathematics means something quite specific. In fact, a lot of words have a specific meaning in maths.

Thanks for the reply but I havnt mistaken “and” for “or” and so I’m still confused
Q:
https://imgur.com/a/URgERid

MS:
https://imgur.com/a/TBJDfpp

Reply 3

tonyiptony

Original post by rhyso4

Thanks for the reply but I havnt mistaken “and” for “or” and so I’m still confused
Q:
https://imgur.com/a/URgERid

MS:
https://imgur.com/a/TBJDfpp

You've written:
"x < -2 , x > 3 , and x > 2.5"
I would ask "what does the comma mean?"
So in a way we do really want to write things accurately.

But I assume you have the right intention of ' "x < -2 or x > 3" and "x > 2.5" ' (note where I put the single and double quotation marks - they are important). If you were to draw your solution to the two inequalities separately on the same real number line (hint, using different colors help), what would you get? Then "and" means we need both to be true, i.e. where both solutions intersect, so...

(edited 6 months ago)

Reply 4

Phys&Math_Ethan

x^2 - x - 6 > 0 and 10 - 2x < 5

So when is your quadratic greater than zero? (x<-2 or x>3)
AND (meaning that the following must also be true)
10-2x<5 (x>5/2)

So x must satisfy both conditions. x cannot be <-2 and >2.5 hence the -ve region is out. Thus it can only be x>3

As stated above, a sketch of both regions on a graph would make it very clear.

Reply 5

ameliasutherland

the question is asking you to find the range of values of x that are true for both of the equation. your critical values are correct, however you have not correctly interpreted what the question is asking of you. for the value of x to satisfy both equations that means that x must be greater than 2.5 AND either less than -2 or greater than 3. if you were to draw these inequalities on a. number line, you would see that the only values of x which satisfy BOTH of these criteria at once are those greater than 3 as that is where both of the lines on the number line would be, therefore your answer must be x>3 :smile: