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09 Jul 2019, 01:55
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C
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How many integer values of x satisfy the equation |x-3|+|x-20|<17?
A. 2
B. 4
C. 0
D. 1
E. 3
A. 2
B. 4
C. 0
D. 1
E. 3
09 Jul 2019, 06:45
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Bookmarks
Kinshook
|x - 3| + |x - 20| signifies sum of 'distance from 3' and 'distance from 20'. Note that distance between 3 and 20 is 17. So whichever point we take on the number line, the sum of the distance will be at least 17. So the sum will never be less than 17.
Answer (C)
Check:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... s-part-ii/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2016/1 ... -part-iii/
14 Oct 2024, 20:46
Kudos
Bookmarks
We need to find How many integer values of x satisfy the equation |x-3|+|x-20|<17?
Let's solve the problem using two methods:
Method 1: Algebra
As we have two absolute values so we will have three cases and the number line will be divided into three points by equating following equations to 0
x - 3 =0 and x - 20 = 0
Method 2: Graph
|x-3| = Distance between x and 3
|x-20| = Distance between x and 20
=> |x-3| + |x-20| = Distance between x and 3 + Distance between x and 20
Now, there can be 5 cases
Case 1: x is smaller than 3
Distance between x and 20 will be greater than the distance between 3 and 20, so will be more than 17
=> Distance between x and 3 + Distance between x and 20 > 17 => it is NOT < 17
case 1.jpg [ 14.5 KiB | Viewed 3567 times ]
Case 2: x = 3
Distance between x and 20 = 17 and distance between x and 3 = 0
=> Distance between x and 3 + Distance between x and 20 = 17 => it is NOT < 17
case 2.jpg [ 10.84 KiB | Viewed 3520 times ]
Case 3: x is between 3 and 20
Distance between x and 3 + Distance between x and 20 = 17 => it is NOT < 17
case-3.jpg [ 10.32 KiB | Viewed 3511 times ]
Case 4: x = 20
Distance between x and 20 = 0 and distance between x and 3 = 17
=> Distance between x and 3 + Distance between x and 20 = 17 => it is NOT < 17
case 4.jpg [ 11.44 KiB | Viewed 3496 times ]
Case 5: x is greater than 20
Distance between x and 3 will be greater than the distance between 3 and 20, so will be more than 17
=> Distance between x and 3 + Distance between x and 20 > 17 => it is NOT < 17
case-5.jpg [ 11.65 KiB | Viewed 3470 times ]
So, Answer will be C
Hope it helps!
Watch the following video to MASTER Inequality + Absolute value Problems
Let's solve the problem using two methods:
Method 1: Algebra
As we have two absolute values so we will have three cases and the number line will be divided into three points by equating following equations to 0
x - 3 =0 and x - 20 = 0
| Number line is divided into three parts x ≤ 3, 3 ≤ x ≤ 20, x ≥ 20 | ||
| -Case 1: x ≤ 3 Now when x ≤ 3 then both x - 3 and x - 20 will be non-positive => | x-3| = -(x-3) and |x-20| = -(x-20) => -(x-3) - (x-20) < 17 => -2x + 23 < 17 => 2x > 23 - 17 => x > 6/2 => x > 3 But the range was x ≤ 3 => NO SOLUTION in this case | -Case 2: 3 ≤ x ≤ 20 Now when 3 ≤ x ≤ 20 then x - 3 will be non-negative and x - 20 will be non-positive => | x-3| = (x-3) and |x-20| = -(x-20) => (x-3) - (x-20) < 17 => 17 < 17 => NO SOLUTION in this case | Case 3: x ≥ 20 Now when x ≥ 20 then both x - 3 and x - 20 will be non-negative => | x-3| = (x-3) and |x-20| = (x-20) => (x-3) + (x-20) < 17 => 2x - 23 < 17 => 2x < 40 => x < 40/2 => x < 20 But the range was x ≥ 20 => NO SOLUTION in this case |
Method 2: Graph
|x-3| = Distance between x and 3
|x-20| = Distance between x and 20
=> |x-3| + |x-20| = Distance between x and 3 + Distance between x and 20
Now, there can be 5 cases
Case 1: x is smaller than 3
Distance between x and 20 will be greater than the distance between 3 and 20, so will be more than 17
=> Distance between x and 3 + Distance between x and 20 > 17 => it is NOT < 17
Attachment:
case 1.jpg [ 14.5 KiB | Viewed 3567 times ]
Case 2: x = 3
Distance between x and 20 = 17 and distance between x and 3 = 0
=> Distance between x and 3 + Distance between x and 20 = 17 => it is NOT < 17
Attachment:
case 2.jpg [ 10.84 KiB | Viewed 3520 times ]
Case 3: x is between 3 and 20
Distance between x and 3 + Distance between x and 20 = 17 => it is NOT < 17
Attachment:
case-3.jpg [ 10.32 KiB | Viewed 3511 times ]
Case 4: x = 20
Distance between x and 20 = 0 and distance between x and 3 = 17
=> Distance between x and 3 + Distance between x and 20 = 17 => it is NOT < 17
Attachment:
case 4.jpg [ 11.44 KiB | Viewed 3496 times ]
Case 5: x is greater than 20
Distance between x and 3 will be greater than the distance between 3 and 20, so will be more than 17
=> Distance between x and 3 + Distance between x and 20 > 17 => it is NOT < 17
Attachment:
case-5.jpg [ 11.65 KiB | Viewed 3470 times ]
So, Answer will be C
Hope it helps!
Watch the following video to MASTER Inequality + Absolute value Problems
