Depth · one concept from first principles · step 2 of 4
Combine sign cases to get extreme distances
If and , find the sum of the greatest and least possible values of .
Number system
Your answer
How to solve
Strategy Make a Systematic List — First pin down |x|. Then x has two choices and y has two, so there are only four cases: list them all and read off the largest and smallest |y-x|. A number line explains the pattern: |y-x| is the distance between the two points.
1STEP 1
Find the size of x
Since |y|=5, the first condition says |x|-5=11.
|x| = 11 + 5 = 16
So x=16 or x=-16, and y=5 or y=-5.
6.NS.C.7Make A Systematic List2STEP 2
List all four sign cases
Compute |y-x| for every pairing of the possible values.
|5-16| = 11, |5-(-16)| = 21, |-5-16| = 21, |-5-(-16)| = 11
Only two different values appear: 11 and 21.
7.NS.A.1Make A Systematic List3STEP 3
See why on a number line
|y-x| is the distance from y to x: on the same side of 0 it is the difference of their sizes, on opposite sides it is the sum.
16-5 = 11, 16+5 = 21
Same side gives the least distance, opposite sides the greatest.
7.NS.A.1Draw A Diagramis the distance between the points and on the number line, which is the difference of their sizes when they sit on the same side of and the sum of their sizes when they sit on opposite sides.
Why?
is the gap between and on the number line, just as and are how far each sits from .
🧱Absolute value is distance from zero|a| is how far a sits from 0 on the number line, whichever side it is on, so it is never negative and |a - b| is the gap between a and b.
Why?
When lies between the two points, it cuts the stretch from to into a piece of length and a piece of length , so the gap is .
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
When both points sit on the same side, the stretch of from to is the stretch of from to plus the gap, so the gap is .
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
Add the greatest and least values
The greatest value is 21 and the least is 11.
21 + 11 = 32
The requested sum is 32.
7.NS.A.1Make A Systematic ListAnswer
32
Try x=-16, y=5: |5-(-16)|=21. Try x=16, y=5: |5-16|=11. No case can be farther apart than 16+5 or closer than 16-5, so 21 and 11 are the extremes and 21+11=32.
Takeaway
Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.
- Find the size of x
- List all four sign cases
- See why on a number line
- Add the greatest and least values
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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