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number of solutions

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by mathbyvemuri » Mon Jun 04, 2012 10:15 pm
7 is the reference point
for x=7, |x-3|+|x-4| = 4+3 = 7
And the inequality is valid for x = 7,6,5,4,3,2,1,0
for x = 0, |x-3|+|x-4| = 3+4 = 7
0 and 7 are the border values of the solution set.
Answer "C"
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by Anurag@Gurome » Mon Jun 04, 2012 10:28 pm
hey_thr67 wrote:The number of integral values of x, that satisfy the inequation |x-3|+|x-4| <= 7 is
Algebraic Approach:
The critical points for the given inequality are x = 3 and x = 4.
Hence, we need to analyze the inequality in the following three regions,

For x < 3
  • |x - 3| = -(x - 3) = (3 - x) and |x - 4| = -(x - 4) = (4 - x)
    --> |x - 3| + |x - 4| ≤ 7
    --> (3 - x) + (4 - x) ≤ 7
    --> (7 - 2x) ≤ 7
    --> x ≥ 0

    Hence, 0 ≤ x < 3 ---> Three integral solutions : 0, 1, and 2
For 3 ≤ x < 4
  • As we can see that only integral solution in this region is x = 3, which does satisfy the inequality we don't need to analyze this in detail.

    Hence, only one integral solutions : 3
For x ≥ 4
  • |x - 3| = (x - 3) and |x - 4| = (x - 4)
    --> |x - 3| + |x - 4| ≤ 7
    --> (x - 3) + (x - 4) ≤ 7
    --> (2x - 7) ≤ 7
    --> 2x ≤ 14
    --> x ≤ 7

    Hence, 4 ≤ x ≤ 7 ---> Four integral solutions : 4, 5, 6, and 7
Therefore, a total of (3 + 1 + 4) = 8 integral solutions.

The correct answer is C.
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by Anurag@Gurome » Mon Jun 04, 2012 10:45 pm
hey_thr67 wrote:The number of integral values of x, that satisfy the inequation |x-3|+|x-4| <= 7 is
Conceptual Approach:
If visualize this problem on the number line, then the problem is saying that sum of the distances of x from 3 and 4 on the number line is less than 7. Let's draw the number line as follows...

Image

Now, from the above figure we can see that, if x is less than zero or greater than 7, then the sum of the distances of x from 3 and 4 will be more than 7. Hence, x must lie between 0 and 7, both inclusive.

Therefore, integral solutions for x are : 0, 1, 2, 3, 4, 5, 6, and 7.

The correct answer is C.
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by hey_thr67 » Tue Jun 05, 2012 9:15 am
I chose visualization but I often get confused with 0. Is it counted as integral number or rational number ?
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by Stuart@KaplanGMAT » Tue Jun 05, 2012 10:25 am
hey_thr67 wrote:I chose visualization but I often get confused with 0. Is it counted as integral number or rational number ?
Both!

0 is an uncharged (i.e. neither positive nor negative) even integer.

As an aside, all integers are rational (but not the other way around).
Image

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by diebeatsthegmat » Tue Jun 05, 2012 10:55 pm
hey_thr67 wrote:The number of integral values of x, that satisfy the inequation |x-3|+|x-4| <= 7is ,

A: 7
B: 6
C: 8
D: 9
E: 10
i dont understand... my answer is A and B, we have to find x which when replaced in the inequation give the resuld <=7
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by Anurag@Gurome » Tue Jun 05, 2012 11:06 pm
diebeatsthegmat wrote:i dont understand... my answer is A and B, we have to find x which when replaced in the inequation give the resuld <=7
The question asks for number of integral values of x not what are the integral values of x...

As I have shown in earlier posts, the possible values of x are : 0, 1, 2, 3, 4, 5, 6, and 7. A total of eight values.

Hope that helps.
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