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

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ZeHanzBest ResponseYou've already chosen the best response.0
First, 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.
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
Don't be! Can you factor x²4x12=0?
 one year ago

gjhfdfgBest ResponseYou've already chosen the best response.0
x*x  2*2*x2*2*3 Like that?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
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?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
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: x²4x12=0 is the same as (x+2)(x6)=0. Are you familiar with this process?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
OK, 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?
 one year ago

gjhfdfgBest ResponseYou've already chosen the best response.0
That the product is 0 still? I dont know..
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
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?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
So suppose I have multiplied two numbers and the outcome IS zero, what do you know about (at least one of) these numbers?
 one year ago

gjhfdfgBest ResponseYou've already chosen the best response.0
I dont get what your asking for, it would just be zero
 one year ago

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

ZeHanzBest ResponseYou've already chosen the best response.0
Can you solve the equation x+2=0?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
OK, 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
 one year ago

gjhfdfgBest ResponseYou've already chosen the best response.0
wouldnt it be 6? instead of just 6
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
No, 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
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
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.
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
To 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:
 one year ago

gjhfdfgBest ResponseYou've already chosen the best response.0
Wouldnt that make them infinitys?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
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:
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
Only one thing left: we have to write the solution (drawn in red) in interval notation...
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
How do you write all the numbers from 2 to 6 (including 2 and 6!) as an interval?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
That would exclude 2 and 6...
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
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...
 one year ago
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