number of solutions

[hey_thr67]

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

[mathbyvemuri]

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"

[Anurag@Gurome]

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.

[Anurag@Gurome]

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...



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.

[hey_thr67]

I chose visualization but I often get confused with 0. Is it counted as integral number or rational number ?

[Stuart@KaplanGMAT]

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).

[diebeatsthegmat]

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

[Anurag@Gurome]

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.