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Negative Numbers and the Number Line: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 The parts bin

A workshop records extra parts with a positive number and missing parts with a negative number. Zero means it has exactly the number of parts needed. A bin is missing 17 parts. What number should the workshop record?
1. Hint 1
  The size of the number records how many parts are involved; its sign records the situation.
2. Hint 2
  Decide which side of the workshop's zero reference represents a shortage.
Answer

$-17$ .

Full solution

The bin has a shortage, so its entry belongs on the negative side of zero.
The size of the shortage is 17 parts, so the workshop records $-17$ .
$-17 < 0$

Reading the entry back gives 17 missing parts, which matches the description.

Answer

$-17$ .

Key idea

A signed record uses the size for the amount and the sign for its relation to a reference.
Problem 1: Mark complete
Problem 2 A printed strip

A strip of a number line shows seven integer labels from left to right, with no integer skipped between neighbors. The middle label is $-2$ . Write all seven labels in order.
1. Hint 1
  Neighboring labels on the strip are exactly one unit apart.
2. Hint 2
  Put the given label in the fourth position, then count three positions to its left and three to its right.
Answer

$-5,\;-4,\;-3,\;-2,\;-1,\;0,\;1$ .

Full solution

There are three positions on each side of the middle label.
Counting left from $-2$ gives $-3$ , then $-4$ , then $-5$ .
Counting right gives $-1$ , then $0$ , then $1$ .

Read the strip from its left end.
$-5,\;-4,\;-3,\;-2,\;-1,\;0,\;1$

Check that there are seven labels, the fourth is $-2$ , and every neighboring pair is one unit apart.
All are integers: negative integers, zero, and a positive integer.

Answer

$-5,\;-4,\;-3,\;-2,\;-1,\;0,\;1$ .

Key idea

Neighboring integers continue in one-unit steps through negative numbers, zero and positive numbers.
Problem 2: Mark complete
Problem 3 Reading the notation

What number does $-(-(-12))$ represent?
1. Hint 1
  A minus sign outside parentheses asks for the opposite of the number inside them.
2. Hint 2
  First settle the inner expression $-(-12)$ , then apply the outer minus sign to that result.
Answer

$-12$ .

Full solution

The number $-12$ lies twelve steps left of zero.
Taking its opposite puts it twelve steps right of zero.
$-(-12) = 12$

The outer minus sign takes the opposite once more, returning to the negative side.
$-(-(-12)) = -12$

Two successive changes to the opposite return the starting number, which checks the result.

Answer

$-12$ .

Key idea

Taking the opposite twice returns the original number.
Problem 3: Mark complete
Problem 4 Posts along a track

Posts stand along a straight track, two meters apart. One post marks position zero. Positions to its right are positive and positions to its left are negative. Position labels give meters from zero.

Not counting the zero post, post A is the fifth post to its left, and post B is the first post to its right. Give the position of each post and the distance between them.
1. Hint 1
  Separate how many post gaps there are from how many meters each gap represents.
2. Hint 2
  Find the distance from zero to each marked post, then choose its sign from its side of zero.
3. Hint 3
  The route from A to B passes through zero, so count the gaps on both parts of that route.
Answer

A: $-10$ meters; B: $2$ meters (or $+2$ meters); distance: 12 meters.

Full solution

Five gaps, each two meters long, put A ten meters to the left of zero.
$5 \times 2 = 10$
Its position is $-10$ meters.
B is one gap to the right of zero, so its position is $2$ meters.

There are five gaps from A to zero and one more from zero to B.
$5 + 1 = 6$
Six gaps of two meters each put the posts 12 meters apart.
$6 \times 2 = 12$

Checking by the two distances from zero gives 12 meters again.
$10 + 2 = 12$

Answer

A: $-10$ meters; B: $2$ meters (or $+2$ meters); distance: 12 meters.

Key idea

Position needs a side of zero, while distance counts the lengths of the gaps traveled.
Problem 4: Mark complete
Problem 5 Two storage rooms

Room A is 11 degrees Celsius below zero, and room B is 4 degrees Celsius below zero. A sealed container may be kept in a room only if the room's temperature is warmer than $-8$ degrees Celsius.

Write the temperature of each room as a signed number and identify every room that meets the requirement.
1. Hint 1
  Translate each description relative to zero before judging the storage requirement.
2. Hint 2
  A warmer temperature has a position farther right on the number line.
3. Hint 3
  Locate each room's reading relative to $-8$ .
Answer

A: $-11$ degrees Celsius; B: $-4$ degrees Celsius; only room B meets the requirement.

Full solution

Both readings are below zero, so A is $-11$ degrees Celsius and B is $-4$ degrees Celsius.

The point $-11$ is farther left than $-8$ .
$-11 < -8$
Room A is colder than the required boundary, so it does not qualify.

The point $-4$ is farther right than $-8$ .
$-4 > -8$
Room B is warmer than the boundary, so B is the only qualifying room.

Answer

A: $-11$ degrees Celsius; B: $-4$ degrees Celsius; only room B meets the requirement.

Key idea

A temperature requirement is tested by comparing signed readings on the same scale.
Problem 5: Mark complete
Problem 6 Sorting a list

Sort the numbers $-6$ , $2.5$ , $0$ , $-\tfrac{1}{2}$ , $7$ and $-21$ into three groups: positive integers, negative integers, and numbers that are not integers. Does every number fit exactly one of the three groups? Explain.
1. Hint 1
  An integer sits exactly on a tick of the number line, and which side of zero it lies on, if either, gives its sign.
2. Hint 2
  For each number, check its side of zero and whether it lands on a tick or between two ticks.
3. Hint 3
  Check whether any number lands on the tick that divides the two sides.
Answer

Positive integers: $7$ . Negative integers: $-6$ and $-21$ . Not integers: $2.5$ and $-\tfrac{1}{2}$ . No: $0$ fits none of the groups, because it is an integer that is neither positive nor negative.

Full solution

The integers are the whole numbers together with their negatives, and they are exactly the numbers that land on the ticks of the number line.

The numbers $7$ , $-6$ and $-21$ land on ticks, so they are integers.
The number $7$ lies right of zero, so it is a positive integer.
The numbers $-6$ and $-21$ lie left of zero, so they are negative integers.

The number $2.5$ lies between the neighboring ticks $2$ and $3$ , so it is not an integer.
The number $-\tfrac{1}{2}$ lies between the neighboring ticks $-1$ and $0$ , so it is not an integer either.

Zero lands on a tick, so it is an integer.
Zero is neither positive nor negative, so it belongs to none of the three groups.

As a check, the six numbers split into one positive integer, two negative integers, two non-integers and zero, which accounts for all six.

Answer

Positive integers: $7$ . Negative integers: $-6$ and $-21$ . Not integers: $2.5$ and $-\tfrac{1}{2}$ . No: $0$ fits none of the groups, because it is an integer that is neither positive nor negative.

Key idea

An integer lands on a tick, its side of zero gives its sign, and zero is an integer with no sign.
Problem 6: Mark complete
Problem 7 Three name cards

Three cards, A, B and C, are placed at the positions $-22$ , $-13$ and $-16$ on a number line in some order, one card at each position. A is to the left of B, and C is to the right of B. What is the position of each card?
1. Hint 1
  The two clues determine which card comes first, which is in the middle, and which comes last.
2. Hint 2
  Place the three negative positions from left to right by counting how far each lies to the left of zero.
3. Hint 3
  Match the left, middle and right positions to the order required by the clues.
Answer

A: $-22$ ; B: $-16$ ; C: $-13$ .

Full solution

A is left of B, and C is right of B, so B must occupy the middle position, with A first and C last.

Of the three positions, $-22$ is farthest left.
The point $-16$ lies between it and $-13$ .
$-22 < -16$ $-16 < -13$
Thus A is at $-22$ , B is at $-16$ , and C is at $-13$ .

These placements put A to the left of B and C to its right, so both clues are satisfied.

Answer

A: $-22$ ; B: $-16$ ; C: $-13$ .

Key idea

Relative-position clues can be matched to the left-to-right order of integer labels.
Problem 7: Mark complete
Problem 8 Sam's record

Sam reads the labels $-11$ , $-10$ , $-9$ , $-8$ , $-7$ while moving along a number line. He reports that going from the first label to the last takes five one-unit steps. Is his report correct? Explain, and state how many one-unit steps the trip takes.
1. Hint 1
  A step moves between neighboring positions.
2. Hint 2
  Count a move when passing from one listed label to the next, and consider whether the starting label is itself a move.
Answer

No; 4 one-unit steps.

Full solution

The first label is the starting position.
The moves are from $-11$ to $-10$ , then to $-9$ , then to $-8$ , then to $-7$ .
There are four moves.

Sam counted five positions rather than the gaps joining them.
The number of gaps is
$5 - 1 = 4$
so the trip takes 4 one-unit steps.

Reading the same labels in reverse also requires four moves, confirming the distance.

Answer

No; 4 one-unit steps.

Key idea

The number of one-unit steps is the number of gaps crossed, not the number of labels visited.
Problem 8: Mark complete
Problem 9 Cards for a display

A display must show each different number exactly once. A student prepares six cards labeled $8$ , $-8$ , $3$ , $-3$ , $0$ and $-0$ . Does the display need all six cards? Give the number of different numbers represented and explain.
1. Hint 1
  Different written forms can mark the same point on a number line.
2. Hint 2
  Consider where taking the opposite sends each number, including a number located at the reference point itself.
Answer

No; 5 different numbers.

Full solution

The labels $8$ and $-8$ represent two different points on opposite sides of zero.
The same is true of $3$ and $-3$ , giving four different nonzero numbers.

Zero is at the point where the two sides meet.
Taking its opposite leaves it in place.
$-0 = 0$
The last two cards therefore represent one number, not two.

The total number of different numbers is
$4 + 1 = 5$
so the display uses either the $0$ card or the $-0$ card, together with the four nonzero cards.

Answer

No; 5 different numbers.

Key idea

Zero is its own opposite because its position stays fixed when the number line is reflected across zero.
Problem 9: Mark complete
Problem 10 Nora's claim

Nora says that between any two different negative integers there is another integer, strictly between them. Is her claim true? Explain your decision.
1. Hint 1
  A claim about every pair must hold even when the chosen integer ticks are very close together.
2. Hint 2
  Consider two neighboring ticks on the negative side of the number line.
Answer

False; for example, $-32$ and $-31$ (any two neighboring negative integers work).

Full solution

Choose the negative integers $-32$ and $-31$ .
They are different and satisfy
$-32 < -31$
They occupy neighboring integer ticks, one step apart.

There is no integer tick strictly between these two positions.
This pair makes Nora's claim false.
Any two neighboring negative integers give the same kind of example.

Answer

False; for example, $-32$ and $-31$ (any two neighboring negative integers work).

Key idea

Neighboring integer ticks have no integer between them, including on the negative side of zero.
Problem 10: Mark complete

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