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Good Inequality with absolute value question

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Good Inequality with absolute value question [#permalink] New post 22 May 2011, 19:32
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50 = 3a + 2b

7 > |–a|

If a and b are both integers, how many possible solutions are there to the system above?

(a) 4
(b) 5
(c) 6
(d) 7
(e) 8
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Re: Good Inequality with absolute value question [#permalink] New post 22 May 2011, 21:10
50 = 3a + 2b
Since the sum of 3a and 2b is to be even (50) either both are odd or both are even. since 2b cant be odd, 3a and 2b should both be even. for 3a to be even, a can be : 2,4,6,8, and 0.
however,
7 > |–a|

This means that -7<a<7
If a is positive/zero: it can have values 0,2,4 and 6. cant be 8. 4 solutions.

But we have to consider negative values for a too. Similar to above negative value of 3a will mean subtracting a number from 2a. for subtracting from an even number (2a) to result in 50, the product 3a has to end with an even digit. so a can be -6,-4,-2.

e.g. if a = -4, 3a= -12 and b= 31

we have 7 solutions uptil now, considering all possible values that 2 can take within this bracket of -7<a<7 and a and b being integers.

Note: We dont have to think of values of b, since a is the one with constraints. if b was -ve, a will have to be positive and a much higher valuse than 7, so these are not to be considerd.
Answer is 7 possible solutions.
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Re: Good Inequality with absolute value question [#permalink] New post 23 May 2011, 17:35
Expert's post
u0422811 wrote:
50 = 3a + 2b

7 > |–a|

If a and b are both integers, how many possible solutions are there to the system above?

(a) 4
(b) 5
(c) 6
(d) 7
(e) 8


Remember that |–a| = |a|
e.g. |–3| = 3 = |3|
If |a| < 7, this implies that the absolute value of a is less than 7. So a could range from -6 to 6 (since a can only be an integer).
Those are 13 values.
Now look at this: 50 = 3a + 2b
50 - 3a = 2b
(50 - 3a)/2 = b
Since b has to be integer too, 50 - 3a must be divisible by 2. Since 50 is even, a should be even too (Even - Even = Even).
From -6 to 6, there are 7 even numbers and 6 odd numbers.
So there are 7 possible solutions (a = -6, b = 34; a = -4, b = 31 etc)
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Re: Good Inequality with absolute value question [#permalink] New post 25 May 2011, 21:12
possible values -7<a<7
for all even 'a's there is an integer value for b.
thus the solution points are 0, +|- 2,4,6.

hence 7. D
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Re: Good Inequality with absolute value question   [#permalink] 25 May 2011, 21:12
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