Practice · Combine sign cases to get extreme distances · Sensim Math

Variant 1 easy

If $|x|-|y|=6$ and $|y|=2$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=6$
$|y|=2$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=2

, the first condition says

|x|-2=6

.

|x| = 6 + 2 = 8

So

x=8

or

x=-8

, and

y=2

or

y=-2

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|2-8| = 6,\quad |2-(-8)| = 10,\quad |-2-8| = 10,\quad |-2-(-8)| = 6

Only two different values appear:

6

and

10

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

8-2 = 6,\quad 8+2 = 10

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

10

and the least is

6

.

10 + 6 = 16

The requested sum is

16

.

Answer: 16

4 · Reviewdoes it hold up?

Try $x=-8$ , $y=2$ : $|2-(-8)|=10$ . Try $x=8$ , $y=2$ : $|2-8|=6$ . No case can be farther apart than $8+2$ or closer than $8-2$ , so $10$ and $6$ are the extremes and $10+6=16$ .

Another way: Draw a number line with marks at $\pm 2$ and $\pm 8$ and measure the four possible gaps directly; the longest and shortest are $10$ and $6$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=8$ and $|y|=2$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 2 easy

If $|x|-|y|=7$ and $|y|=3$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=7$
$|y|=3$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=3

, the first condition says

|x|-3=7

.

|x| = 7 + 3 = 10

So

x=10

or

x=-10

, and

y=3

or

y=-3

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|3-10| = 7,\quad |3-(-10)| = 13,\quad |-3-10| = 13,\quad |-3-(-10)| = 7

Only two different values appear:

7

and

13

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

10-3 = 7,\quad 10+3 = 13

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

13

and the least is

7

.

13 + 7 = 20

The requested sum is

20

.

Answer: 20

4 · Reviewdoes it hold up?

Try $x=-10$ , $y=3$ : $|3-(-10)|=13$ . Try $x=10$ , $y=3$ : $|3-10|=7$ . No case can be farther apart than $10+3$ or closer than $10-3$ , so $13$ and $7$ are the extremes and $13+7=20$ .

Another way: Draw a number line with marks at $\pm 3$ and $\pm 10$ and measure the four possible gaps directly; the longest and shortest are $13$ and $7$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=10$ and $|y|=3$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 3 easy

If $|x|-|y|=5$ and $|y|=8$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=5$
$|y|=8$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=8

, the first condition says

|x|-8=5

.

|x| = 5 + 8 = 13

So

x=13

or

x=-13

, and

y=8

or

y=-8

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|8-13| = 5,\quad |8-(-13)| = 21,\quad |-8-13| = 21,\quad |-8-(-13)| = 5

Only two different values appear:

5

and

21

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

13-8 = 5,\quad 13+8 = 21

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

21

and the least is

5

.

21 + 5 = 26

The requested sum is

26

.

Answer: 26

4 · Reviewdoes it hold up?

Try $x=-13$ , $y=8$ : $|8-(-13)|=21$ . Try $x=13$ , $y=8$ : $|8-13|=5$ . No case can be farther apart than $13+8$ or closer than $13-8$ , so $21$ and $5$ are the extremes and $21+5=26$ .

Another way: Draw a number line with marks at $\pm 8$ and $\pm 13$ and measure the four possible gaps directly; the longest and shortest are $21$ and $5$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=13$ and $|y|=8$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 4 medium

If $|x|-|y|=8$ and $|y|=9$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=8$
$|y|=9$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=9

, the first condition says

|x|-9=8

.

|x| = 8 + 9 = 17

So

x=17

or

x=-17

, and

y=9

or

y=-9

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|9-17| = 8,\quad |9-(-17)| = 26,\quad |-9-17| = 26,\quad |-9-(-17)| = 8

Only two different values appear:

8

and

26

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

17-9 = 8,\quad 17+9 = 26

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

26

and the least is

8

.

26 + 8 = 34

The requested sum is

34

.

Answer: 34

4 · Reviewdoes it hold up?

Try $x=-17$ , $y=9$ : $|9-(-17)|=26$ . Try $x=17$ , $y=9$ : $|9-17|=8$ . No case can be farther apart than $17+9$ or closer than $17-9$ , so $26$ and $8$ are the extremes and $26+8=34$ .

Another way: Draw a number line with marks at $\pm 9$ and $\pm 17$ and measure the four possible gaps directly; the longest and shortest are $26$ and $8$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=17$ and $|y|=9$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 5 easy

If $|x|-|y|=9$ and $|y|=4$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=9$
$|y|=4$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=4

, the first condition says

|x|-4=9

.

|x| = 9 + 4 = 13

So

x=13

or

x=-13

, and

y=4

or

y=-4

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|4-13| = 9,\quad |4-(-13)| = 17,\quad |-4-13| = 17,\quad |-4-(-13)| = 9

Only two different values appear:

9

and

17

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

13-4 = 9,\quad 13+4 = 17

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

17

and the least is

9

.

17 + 9 = 26

The requested sum is

26

.

Answer: 26

4 · Reviewdoes it hold up?

Try $x=-13$ , $y=4$ : $|4-(-13)|=17$ . Try $x=13$ , $y=4$ : $|4-13|=9$ . No case can be farther apart than $13+4$ or closer than $13-4$ , so $17$ and $9$ are the extremes and $17+9=26$ .

Another way: Draw a number line with marks at $\pm 4$ and $\pm 13$ and measure the four possible gaps directly; the longest and shortest are $17$ and $9$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=13$ and $|y|=4$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 6 medium

If $|x|-|y|=4$ and $|y|=10$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=4$
$|y|=10$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=10

, the first condition says

|x|-10=4

.

|x| = 4 + 10 = 14

So

x=14

or

x=-14

, and

y=10

or

y=-10

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|10-14| = 4,\quad |10-(-14)| = 24,\quad |-10-14| = 24,\quad |-10-(-14)| = 4

Only two different values appear:

4

and

24

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

14-10 = 4,\quad 14+10 = 24

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

24

and the least is

4

.

24 + 4 = 28

The requested sum is

28

.

Answer: 28

4 · Reviewdoes it hold up?

Try $x=-14$ , $y=10$ : $|10-(-14)|=24$ . Try $x=14$ , $y=10$ : $|10-14|=4$ . No case can be farther apart than $14+10$ or closer than $14-10$ , so $24$ and $4$ are the extremes and $24+4=28$ .

Another way: Draw a number line with marks at $\pm 10$ and $\pm 14$ and measure the four possible gaps directly; the longest and shortest are $24$ and $4$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=14$ and $|y|=10$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 7 medium

If $|x|-|y|=11$ and $|y|=5$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=11$
$|y|=5$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

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

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|5-16| = 11,\quad |5-(-16)| = 21,\quad |-5-16| = 21,\quad |-5-(-16)| = 11

Only two different values appear:

11

and

21

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

16-5 = 11,\quad 16+5 = 21

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

21

and the least is

11

.

21 + 11 = 32

The requested sum is

32

.

Answer: 32

4 · Reviewdoes it hold up?

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 $21+11=32$ .

Another way: Draw a number line with marks at $\pm 5$ and $\pm 16$ and measure the four possible gaps directly; the longest and shortest are $21$ and $11$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=16$ and $|y|=5$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 8 medium

If $|x|-|y|=12$ and $|y|=3$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=12$
$|y|=3$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=3

, the first condition says

|x|-3=12

.

|x| = 12 + 3 = 15

So

x=15

or

x=-15

, and

y=3

or

y=-3

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|3-15| = 12,\quad |3-(-15)| = 18,\quad |-3-15| = 18,\quad |-3-(-15)| = 12

Only two different values appear:

12

and

18

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

15-3 = 12,\quad 15+3 = 18

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

18

and the least is

12

.

18 + 12 = 30

The requested sum is

30

.

Answer: 30

4 · Reviewdoes it hold up?

Try $x=-15$ , $y=3$ : $|3-(-15)|=18$ . Try $x=15$ , $y=3$ : $|3-15|=12$ . No case can be farther apart than $15+3$ or closer than $15-3$ , so $18$ and $12$ are the extremes and $18+12=30$ .

Another way: Draw a number line with marks at $\pm 3$ and $\pm 15$ and measure the four possible gaps directly; the longest and shortest are $18$ and $12$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=15$ and $|y|=3$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 9 hard

If $|x|-|y|=10$ and $|y|=12$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=10$
$|y|=12$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=12

, the first condition says

|x|-12=10

.

|x| = 10 + 12 = 22

So

x=22

or

x=-22

, and

y=12

or

y=-12

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|12-22| = 10,\quad |12-(-22)| = 34,\quad |-12-22| = 34,\quad |-12-(-22)| = 10

Only two different values appear:

10

and

34

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

22-12 = 10,\quad 22+12 = 34

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

34

and the least is

10

.

34 + 10 = 44

The requested sum is

44

.

Answer: 44

4 · Reviewdoes it hold up?

Try $x=-22$ , $y=12$ : $|12-(-22)|=34$ . Try $x=22$ , $y=12$ : $|12-22|=10$ . No case can be farther apart than $22+12$ or closer than $22-12$ , so $34$ and $10$ are the extremes and $34+10=44$ .

Another way: Draw a number line with marks at $\pm 12$ and $\pm 22$ and measure the four possible gaps directly; the longest and shortest are $34$ and $10$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=22$ and $|y|=12$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 10 hard

If $|x|-|y|=13$ and $|y|=6$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=13$
$|y|=6$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=6

, the first condition says

|x|-6=13

.

|x| = 13 + 6 = 19

So

x=19

or

x=-19

, and

y=6

or

y=-6

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|6-19| = 13,\quad |6-(-19)| = 25,\quad |-6-19| = 25,\quad |-6-(-19)| = 13

Only two different values appear:

13

and

25

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

19-6 = 13,\quad 19+6 = 25

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

25

and the least is

13

.

25 + 13 = 38

The requested sum is

38

.

Answer: 38

4 · Reviewdoes it hold up?

Try $x=-19$ , $y=6$ : $|6-(-19)|=25$ . Try $x=19$ , $y=6$ : $|6-19|=13$ . No case can be farther apart than $19+6$ or closer than $19-6$ , so $25$ and $13$ are the extremes and $25+13=38$ .

Another way: Draw a number line with marks at $\pm 6$ and $\pm 19$ and measure the four possible gaps directly; the longest and shortest are $25$ and $13$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=19$ and $|y|=6$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 11 hard

If $|x|-|y|=15$ and $|y|=7$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=15$
$|y|=7$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=7

, the first condition says

|x|-7=15

.

|x| = 15 + 7 = 22

So

x=22

or

x=-22

, and

y=7

or

y=-7

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|7-22| = 15,\quad |7-(-22)| = 29,\quad |-7-22| = 29,\quad |-7-(-22)| = 15

Only two different values appear:

15

and

29

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

22-7 = 15,\quad 22+7 = 29

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

29

and the least is

15

.

29 + 15 = 44

The requested sum is

44

.

Answer: 44

4 · Reviewdoes it hold up?

Try $x=-22$ , $y=7$ : $|7-(-22)|=29$ . Try $x=22$ , $y=7$ : $|7-22|=15$ . No case can be farther apart than $22+7$ or closer than $22-7$ , so $29$ and $15$ are the extremes and $29+15=44$ .

Another way: Draw a number line with marks at $\pm 7$ and $\pm 22$ and measure the four possible gaps directly; the longest and shortest are $29$ and $15$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=22$ and $|y|=7$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Variant 12 hard

If $|x|-|y|=17$ and $|y|=6$ , find the sum of the greatest and least possible values of $|y-x|$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know the size of $y$ and how much bigger the size of $x$ is. Each of $x$ and $y$ could be positive or negative, and we want the largest and smallest values that the distance $|y-x|$ can take, then their sum.

Givens

$|x|-|y|=17$
$|y|=6$

Unknowns

The greatest value of $|y-x|$ , the least value of $|y-x|$ , and their sum.

Constraints

An absolute value fixes a number's size but not its sign, so $x$ and $y$ each have two possible values.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

3 · Execute4 carry out the plan

1Find the size of $x$

#2 Make a Systematic List 6.NS.C.7

Since

|y|=6

, the first condition says

|x|-6=17

.

|x| = 17 + 6 = 23

So

x=23

or

x=-23

, and

y=6

or

y=-6

.

2List all four sign cases

#2 Make a Systematic List 7.NS.A.1

Compute

|y-x|

for every pairing of the possible values.

|6-23| = 17,\quad |6-(-23)| = 29,\quad |-6-23| = 29,\quad |-6-(-23)| = 17

Only two different values appear:

17

and

29

.

3See why on a number line

#1 Draw a Diagram 7.NS.A.1

|y-x|

is the distance between the points

y

and

x

. When they are on the same side of

0

the distance is the difference of their sizes; on opposite sides it is the sum.

23-6 = 17,\quad 23+6 = 29

Same side gives the least distance, opposite sides the greatest.

4Add the greatest and least values

#2 Make a Systematic List 7.NS.A.1

The greatest value is

29

and the least is

17

.

29 + 17 = 46

The requested sum is

46

.

Answer: 46

4 · Reviewdoes it hold up?

Try $x=-23$ , $y=6$ : $|6-(-23)|=29$ . Try $x=23$ , $y=6$ : $|6-23|=17$ . No case can be farther apart than $23+6$ or closer than $23-6$ , so $29$ and $17$ are the extremes and $29+17=46$ .

Another way: Draw a number line with marks at $\pm 6$ and $\pm 23$ and measure the four possible gaps directly; the longest and shortest are $29$ and $17$ .

Standardsmin grade 7

6.NS.C.7 Order and absolute value of rational numbers — Reading $|x|=23$ and $|y|=6$ as two possible values each.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Computing $|y-x|$ as the distance between two signed numbers.

💡Takeaway. Absolute value tells you the size; try both signs and the distance shows its biggest and smallest values.

Combine sign cases to get extreme distances · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?