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
number of solutions
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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"
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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Algebraic Approach:hey_thr67 wrote:The number of integral values of x, that satisfy the inequation |x-3|+|x-4| <= 7 is
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
- 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
- |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
The correct answer is C.
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Conceptual Approach:hey_thr67 wrote:The number of integral values of x, that satisfy the inequation |x-3|+|x-4| <= 7 is
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.
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Both!hey_thr67 wrote:I chose visualization but I often get confused with 0. Is it counted as integral number or rational number ?
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).
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i dont understand... my answer is A and B, we have to find x which when replaced in the inequation give the resuld <=7hey_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
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The question asks for number of integral values of x not what are the integral values of x...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
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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