Sign Diagram
It's late, but whatever.
So we have this Inequality here;
x² - 2x - 8 ≥ 0
So we're gonna go around about solving it like this;
STEP 1;
Determine the Zeros of the function and place on a number line.
x² - 2x - 8 ≥ 0 → (x - 4)(x + 2) ≥ 0 → Zeros x = 4, -2
Now as you see this little bugger here, pay attention to the sign → ≥
Based on the sign, you'll know what you're looking for. So for this sign, we're looking for a positive (+) region.
So let's put it on this little dandy number line. The positive areas represents that, those areas are positive, and the negative area represents the area between those two numbers is negative. I'll show you how we figured out which region is positive/negative lower down.
As you notice the circles on this number line, there
is a rule ;> < → OPEN CIRCLE
≥ ≤ → CLOSED CIRCLE
Therefore, depending on the sign, the cirlce will be opened or closed.
STEP 2; Test a value from each region.
So lets test values from each region;
Test region (-∞, -2] So let's go ahead and try it on 4, -2
Lets just throw out a big number like -46545489454498457313216548. I think that's big enough.
So we go along and → -46545489454498457313216548 - 4 = -46545489454498457313216552.
Hey! A negative number.
Now try it on -2.
-46545489454498457313216548 - -2 = -46545489454498457313216546.
Hm, still a negative number. Well with those two negatives, you multiply them to get... ( - x - = +) a positive value. Therefore, you've found the region you're looking for.
Now let's test another region
Test region [-2, 4]
Lets pick a number like -1. Sure.
Testing... → -1 -4 = -5
→ -1 + 2 = 1 (Since it was -2, I didn't want to bother going - -2.)
So now we have ( - x + = -) Nope. Not the region we're looking for.
Let's test the last region now
Test region [4, ∞)
So I don't know, lets pick a number like, 1454589658421259. No, no let's go with something like 5.
Testing... → 5 -4 = 1
→ 5+2 = 7
Hey, so now we have (+ x + = +). Found it. ^_^
THEREFORE, the solutions are (-∞, -2], [4,∞). Or you could do it the LONG interval notation way which would be ; X ≤ -2. X ≥ 4. DONE!
Now let's make it a little scary.
-x² +3x +10 > 0.
Oh no it's a negative sign, rip up your papers now! Oh now wait, there's an easier way than burning up your paper. Just get rid of the negative so you can factor. To do so, just divide every term by - . Just to let you know, every sign will change. Yes including the greater than sign.
→x² -3x -10 < style="color: rgb(255, 0, 0);">NOTE; KNOW THAT THE SIGN HAS CHANGED.
Now we can factor this little bugger. It boils down to (x-5)(x+2) < style="color: rgb(0, 0, 0);">So since the sign is <, we're looking for a negative region.
I think she might have just done that to mess with us, but the truth is you don't have to do any calculations! Just use 0.
So let's try it.
→0-5 = -5
→0+2 = 2
So the interval notation for this is (-2, 5) or you could do it the long way → x > -2, x <> which could easily be done this way → -2< x < style="" face="arial" size="3"> So you could've done it in one whole shot! Therefore it would be negative two is less than x which is less than 5.
Reminder that we have two dreadful Yellow sheets. Now who should be the next scribe... Roxanne? No now, lets go with Skyline! ^_^ No wait. I have a better idea. Let's give the honors to Elven!
No hard feelings? Haha
Note to self; Do not blog using firefox anymore.
Well I know THERE MUST'VE been something wrong in this blog, so somebody please point out a mistake if I have one.. and so on and so forth.
And to end it on a funny note (I hope I don't get introuble for this.);
posted by Rence @ 12:35 PM
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