Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation.

- anonymous

- katieb

See more answers at brainly.com

At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your **free** account and access **expert** answers to this

and **thousands** of other questions

- anonymous

x^2 - 4x - 12 ≤ 0

- ZeHanz

First, solve x²-4x-12=0, by factoring as (x ....)(x ....)=0.
Once you've found the zeroes, put these on the number line.
They will divide the number line into three pieces.
Because the graph of x²-4x-12 is a parabola with a minimum, the part of the number line between the two zeroes gives a negative value, the other parts a positive value.
Mark the areas with "+ + +", "- - -" and "+ + +".
Now you can see the solution of the inequality right in front of you: it is the middle part.

- anonymous

Im lost

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- ZeHanz

Don't be! Can you factor x²-4x-12=0?

- anonymous

x*x - 2*2*x-2*2*3
Like that?

- ZeHanz

No, like (x + ...)(x + ...)=0. On the dots you need numbers a and b.
Now a+b=-4 (that's the coefficient of the middle term -4x),
and a*b = -12, the constant third term.
Can you find a and b?

- anonymous

2 & 12 being a & b?

- ZeHanz

No, because 2+12 = 14 instead of -4 and 2*12 = 24 instead of -12.
If you take a = 2 and b = -6, you get 2+(-6)=-4 and 2*(-6)=-12, so:
x²-4x-12=0 is the same as (x+2)(x-6)=0.
Are you familiar with this process?

- anonymous

Kinda,

- ZeHanz

OK, you probably need to practise this a few times, to get the hang of it.
Now look at (x+2)(x-6)=0. It is the product of two numbers, x+2 and x-6.
This product is 0.
If you multiply two numbers and the outcome is 0, what does that tell you about these numbers?

- anonymous

That the product is 0 still? I dont know..

- ZeHanz

If you have 4*6, 2*59, 63*456, you may not always be able to give the answer immediately, but one thing you do know: the answer is certainly NOT zero. Agreed?

- anonymous

Agreed,

- ZeHanz

So suppose I have multiplied two numbers and the outcome IS zero, what do you know about (at least one of) these numbers?

- anonymous

I dont get what your asking for, it would just be zero

- ZeHanz

It means that at least one of the numbers must be zero!
0*456=0, -34*0=0, 0*0=0, but 34*53 is not 0.
This means, if you have (x+2)(x-6)=0, you can split this equation into two very easy ones:
x+2=0 or x-6=0.
Can you see their solutions?

- anonymous

No

- ZeHanz

Can you solve the equation x+2=0?

- anonymous

-2

- ZeHanz

OK, and in the same way, the solution of x-6=0 is 6.
We now have the solutions of (x+2)(x-6)=0.
Solutions are -2 and 6.
Put them on a number line, like this:
------------|------------|-----------
-2 6

- anonymous

wouldnt it be -6? instead of just 6

- ZeHanz

No, because we need to set x=6 in the equation x-6=0 to make it true.
Next step:
We can also put zeroes above -2 and 6 to indicate the value of (x+2)(x-6) there:
0 0
------------|------------|-----------
-2 6

- ZeHanz

Keep in mind we are trying to solve x² - 4x - 12 ≤ 0.
We have indicated on the number line this: x² - 4x - 12 = 0.
That is not enough, we also need to know what values of x will make the thing < 0.

- ZeHanz

To find out what the sign (pos or neg) x²-4x-12 will get, just try a number between -2 and 6, e.g. x=0.
0² - 4*0 -12 = -12 <0, so negative there.
If you try in the other parts of the line you get this:

##### 1 Attachment

- anonymous

Wouldnt that make them infinitys?

- ZeHanz

If you put in very large numbers for x, the value of x^2 - 4x - 12 will also get very large, but we need not to worry about that, because we have to answer another question:
What values of x make x^2 - 4x - 12 smaller than zero?
The answer is visible in the image: between -2 and 6!
So if I color all solutions red, I get this:

##### 1 Attachment

- ZeHanz

Only one thing left: we have to write the solution (drawn in red) in interval notation...

- ZeHanz

How do you write all the numbers from -2 to 6 (including -2 and 6!) as an interval?

- anonymous

(-2,6)

- ZeHanz

That would exclude -2 and 6...

- ZeHanz

To include them, you write [-2, 6].
One final word of advice: from the answers you've given I get the impression that you are not yet ready to solve problems like this. I really think you should practise elementary algebra skills.
Nevertheless, I hope I have helped you a little...

Looking for something else?

Not the answer you are looking for? Search for more explanations.