Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
Fig can you elaborate on the logic of your answer ( Z level students teaching style plz?
thanks in advance
At least, I can try
Since x(x-5)(x+2)=0, then x could be -2, 0 or 5.
If we have a close look on these possible solutions, we can observe that 1 is negative, 1 is without sign and 1 is positive.
To answer the problem quesion :'is x negative?', we could have these groups of solutions:
> groupe 1 : x = 5 (Not neg)
> groupe 2 : x = 5 or x = 0 (Not neg)
> groupe 3 : x = 0 (Not neg)
> groupe 4 : x = -2 (Neg)
Now, we could look at the statments. These statments have normally to give us 1 of the above group in order to respond all answers except (E).
Stat1: x^2 - 7x >= 0
Thus, x*(x-7) >=0
<=> x <= 0 or x >= 7
The bold inequality shows us that x is negative or 0. It's not match to 1 of the 4 groups definied to answer the question. -2 is negative but 0 is neither negative nor positive.
Stat2: x^2 - 2x - 15>=0
Thus, (x-5)(x+3) >=0
<=> x >= 5 or x <= -3
Bingo, -2 and 0 are out the possible values of x while x=5 (groupe 1) is contained in the bold inequality.
Getting bk to the inequality basics,...i'm awfully weak in inequalities... thus, (x-5)(x+3) >=0 <=> x >= 5 or x <= -3
does x+3 >=0, not mean tht x>= -3( when u subtract 3 on both sides...) please clear this doubt for me.....it wud help me a lot..coz inequality probs always pull me down..
Actually, the equation (x-5)(x+3) = x^2 - 2x - 15 = a*x^2 + b*x + c has the sign of : o a (1 here) when x is not between the 2 roots, thus > 0 o -a (-1) when x is between the 2 roots, thus < 0
By the way, we have so: (x-5)(x+3) >=0 <=> x >= 5 or x <= -3
Fig - Thanks for the explanation. I kinda get it but am still a bit confused. How do you get x<=-3 ...when I subtarct three from x+3>=0 I get x>=-3. Thanks very much in advance.
Well, I can try
To seach (x-5)(x+3) >=0 is similar to study the sign of the function f(x) = (x-5)(x+3).
A great tool that Maths gives us is the table of signs. Once a fonction is factorized in "famous" forms of basic functions, such as a*x+b, we decompose the function and use the properties of the multiplication to determine what values of x give a positive sign and what values of x give a negative sign to the studied function f(x).
This table of signs is fast to draw and brings to the solution
I attached u the resulting table. To find the result signs for f(x), we multiply vertically + and - as they are +1 and -1.
So, when x <= -3 : sign(f(x)) = sign( (x-5)(x+3) ) = (-1)*(-1) = +1
I also join u the draw of the function f(x). U will visually remark that f(x) is positive on values of x not between the 2 roots.
This result is linked to my former post. The sign of a*x^2 + b*x + c
Graph.jpg [ 31.16 KiB | Viewed 629 times ]
Signs-Table.jpg [ 14.17 KiB | Viewed 626 times ]
Last edited by Fig on 11 Oct 2006, 23:49, edited 2 times in total.