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# maths question..inequalties on number line help watch

1. how to you use a number line to see where two sets of inequalties overlap

ok so the question is : write down the range of where both inequalties are true

the inequalties are: x<3 and -2<x<9

NOTE: < means less than or equal too, i couldn't get the proper symbol on keyboard so ive just put less than.

Any help would be most appreciated!
2. Draw the line x=3 on a graph. Everything to the left of that line (including the line itself since this is less than or equal to) is the region of all points where x<3. You can do a similar thing for the second inequality, but you will need to draw two lines.

3. if you combine the inequalities then you will get -2<x<3, because anything above 3 wouldn't be in the range of x<3
4. The way I used to find best to do this is as follows.

(1.) Draw a number line and label the significant numbers (e.g. in this case -2, 3 and 9 are mentioned, so label those)
(2.) For each inequality you're given, shade in the bits which aren't in the region specified by that inequality
(3.) The inequality you should then get is the region which is unshaded

So for example if you had -2<x<5 and 0<x<8, you would:
(1.) Draw a number line, and label -2, 0, 5, 8 on it
(2.) Look at the inequality -2<x<5; you shade everything which is less than 2 and everything which is greater than 5. Now look at the inequality 0<x<8; you shade everything which is less than 0 and everything which is greater than 8.
(3.) You'll notice that you're left with the only unshaded region being between 0 and 5, so "-2<x<5 and 0<x<8" is the same as "0<x<5".
if you combine the inequalities then you will get -2<x<3, because anything above 3 wouldn't be in the range of x<3
how do i combine them?
6. (Original post by nuodai)
The way I used to find best to do this is as follows.

(1.) Draw a number line and label the significant numbers (e.g. in this case -2, 3 and 9 are mentioned, so label those)
(2.) For each inequality you're given, shade in the bits which aren't in the region specified by that inequality
(3.) The inequality you should then get is the region which is unshaded

So for example if you had -2<x<5 and 0<x<8, you would:
(1.) Draw a number line, and label -2, 0, 5, 8 on it
(2.) Look at the inequality -2<x<5; you shade everything which is less than 2 and everything which is greater than 5. Now look at the inequality 0<x<8; you shade everything which is less than 0 and everything which is greater than 8.
(3.) You'll notice that you're left with the only unshaded region being between 0 and 5, so "-2<x<5 and 0<x<8" is the same as "0<x<5".
is there no other way of combining them and getting the same answer?
7. (Original post by cosmicalgebra)
is there no other way of combining them and getting the same answer?
What do you mean?

Essentially what's going on is that each inequality gives you an interval, and when you combine them you just make the intervals shorter so that what you're left with satisfies both inequalities.

So let's look at your example: you have x<3 and -2<x<9. Take x<3, this is just all the numbers less than 3, no matter how much less they are. But then we also have -2<x<9. Now, the x<9 part of this has no bearing, since if x<3 then it's already less than 9. However, we also have -2<x, i.e. x>-2, which means that any number which is less than -2 can't be part of the final interval. As such, what you're left with is numbers which are greater than -2 and less than 3... i.e., -2<x<3.

This is exactly the process followed by shading in parts of the number line, except instead of doing it in your head you're just putting it on paper. It makes it easier to visualise if you've tried doing it on a number line a few times.

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