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

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ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1First, solve x²4x12=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 x²4x12 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.

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1Don't be! Can you factor x²4x12=0?

gjhfdfg
 2 years ago
Best ResponseYou've already chosen the best response.0x*x  2*2*x2*2*3 Like that?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1No, 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?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1No, 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: x²4x12=0 is the same as (x+2)(x6)=0. Are you familiar with this process?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1OK, you probably need to practise this a few times, to get the hang of it. Now look at (x+2)(x6)=0. It is the product of two numbers, x+2 and x6. This product is 0. If you multiply two numbers and the outcome is 0, what does that tell you about these numbers?

gjhfdfg
 2 years ago
Best ResponseYou've already chosen the best response.0That the product is 0 still? I dont know..

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1If 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?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1So suppose I have multiplied two numbers and the outcome IS zero, what do you know about (at least one of) these numbers?

gjhfdfg
 2 years ago
Best ResponseYou've already chosen the best response.0I dont get what your asking for, it would just be zero

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1It 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)(x6)=0, you can split this equation into two very easy ones: x+2=0 or x6=0. Can you see their solutions?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1Can you solve the equation x+2=0?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1OK, and in the same way, the solution of x6=0 is 6. We now have the solutions of (x+2)(x6)=0. Solutions are 2 and 6. Put them on a number line, like this:  2 6

gjhfdfg
 2 years ago
Best ResponseYou've already chosen the best response.0wouldnt it be 6? instead of just 6

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1No, because we need to set x=6 in the equation x6=0 to make it true. Next step: We can also put zeroes above 2 and 6 to indicate the value of (x+2)(x6) there: 0 0  2 6

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1Keep 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
 2 years ago
Best ResponseYou've already chosen the best response.1To find out what the sign (pos or neg) x²4x12 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:

gjhfdfg
 2 years ago
Best ResponseYou've already chosen the best response.0Wouldnt that make them infinitys?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1If 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:

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1Only one thing left: we have to write the solution (drawn in red) in interval notation...

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1How do you write all the numbers from 2 to 6 (including 2 and 6!) as an interval?

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1That would exclude 2 and 6...

ZeHanz
 2 years ago
Best ResponseYou've already chosen the best response.1To 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...
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