Expert: Scott A Wilson - 4/21/2011

Question
Pls help me to solve this Q(1a)solve the inequality 2{x-2}/3-1{x+5}/6 less than or equal to 0. Q(1b):Given that P=xsquare-ysquare/xsquare+xy, (i)Express P in its simplest form; (ii)find the value of P if x=-4 and y=-6.

Answer
As I read them, the questions are sent first and then answered here as well.
Let me know if they are different than this.  NOte that number one has a -1(x+5),
and that is changed to -(x+5), since the 1 multiplied out front can be dropped.

 1a: Inequality 2(x-2)/3 - (x+5)/6 <= 0.
 1b: Given that P = (x^2 - y^2)/(x^2 + xy)
   (i)Express P in its simplest form
   (ii)find the value of P if x=-4 and y=-6.

Note that when an inequality is multiplied by a negative,
less thans become greater thans and vice versa.

1a. This can be multiplied by 6 giving 4(x-2) - (x+5) <= 0.
Multiplying out the parenthsis gives 4x - 8 - x - 5 <= 0.
Combining like terms gives 3x - 13 <= 0.
Adding 13 to both sides gives 3x <= 13.
Dividing both sides by 3 gives x <= 13/3.

Putting this back into 2(x-2)/3 - (x+5)/6 gives us 2(13/3 - 2)/3 - (13/3 + 5)/6.
Mulitplying out gives us 26/9 - 4/3 - 13/18 - 5/6.
Converting all fractions to 18ths gives 52/18 - 24/18 - 13/18 - 15/18.
Combining them gives (52 - 24 - 13 - 15)/18.
Now 24 + 13 + 15 = 52, so this is 52 - 52 in the numerator, and that is 0, so this checks out.

1b. That factors into (x-y)(x+y)/(x(x+y)).
Doing this lets us see that there is an x+y in the numerator and the denominator.
(i) Cancelling this gives P = (x-y)/x.
(ii) Putting in x=-4 and y=-6 gives P (-4 - -6)/(-4).
     That is the same as (-4+6)/(-4) = 2/(-4) = -1/2.