Integers and the Number Line: Chapter Test

20 multiple-choice questions and 10 core practice problems, drawn from across the chapter and mixed together.

Multiple choice

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Question 1 of 20

1

Which statement about the number $0$ is true?

Answer choices for question 1 A $0$ is the smallest positive integer, since every other positive integer stands above it. B $0$ is negative, because it can be written as $-0$ . C $0$ is neither positive nor negative, and it is its own opposite. D $0$ is not an integer, because it counts nothing.
2

Exactly one of these four statements is true. Which one?

Answer choices for question 2 A $-45 > -9$ B $-9 > -45$ C $-45 = 45$ D $0 < -3$
3

Evaluate $-14 - (-9)$ .

Answer choices for question 3 A $-23$ B $5$ C $23$ D $-5$
4

How many integers have an absolute value of $18$ ?

Answer choices for question 4 A Exactly two, namely $18$ and $-18$ . B Exactly one, namely $18$ . C Exactly one, namely $-18$ . D Infinitely many, since every integer at least $18$ units from zero qualifies.
5

Which list is ordered from least to greatest?

Answer choices for question 5 A $-3,\; -17,\; -24,\; 0,\; 8$ B $0,\; -24,\; -17,\; -3,\; 8$ C $-24,\; -17,\; -3,\; 0,\; 8$ D $8,\; 0,\; -3,\; -17,\; -24$
6

Evaluate $(-91) \div (-7)$ .

Answer choices for question 6 A $13$ B $-13$ C $84$ D $637$
7

How far apart are $-17$ and $6$ on the number line?

Answer choices for question 7 A $11$ B $-23$ C $17$ D $23$
8

Exactly one of these four statements is true. Which one?

Answer choices for question 8 A The opposite of $-29$ is less than $-29$ . B The opposite of $-29$ is greater than $0$ . C The opposite of $7$ is greater than $7$ . D The opposite of $0$ is greater than $0$ .
9

Evaluate $-6 + 13 - 21 - (-5)$ .

Answer choices for question 9 A $-19$ B $3$ C $9$ D $-9$
10

Exactly one of these four expressions is negative. Which one?

Answer choices for question 10 A $\lvert -28 \rvert$ B $-(-\lvert -28 \rvert)$ C $-\lvert -28 \rvert$ D $\lvert 28 \rvert$
11

Evaluate $(-4) \times (-6) \times (-2)$ .

Answer choices for question 11 A $-12$ B $-48$ C $48$ D $24$
12

Evaluate $\lvert (-16) + 9 \rvert$ .

Answer choices for question 12 A $7$ B $25$ C $-7$ D $-25$
13

Which of these integers lies between $-9$ and $-4$ on the number line?

Answer choices for question 13 A $-3$ B $-11$ C $7$ D $-7$
14

Exactly one of these four statements is true. Which one?

Answer choices for question 14 A $0 \div (-16)$ is undefined, because a zero cannot take part in a division. B $(-16) \div 0 = 0$ , because the divisor is zero. C $(-16) \times 0 = 0$ , so a zero factor settles a product outright instead of contributing a sign to it. D $0 \div 0 = 1$ , because any number divided by itself is one.
15

Among the integers $-27$ , $-15$ , $4$ and $11$ , which is the greatest, and which lies farthest from zero?

Answer choices for question 15 A $11$ is the greatest, and $-27$ lies farthest from zero. B $11$ is the greatest, and $11$ lies farthest from zero. C $-27$ is the greatest, and $-27$ lies farthest from zero. D $-27$ is the greatest, and $11$ lies farthest from zero.
16

Evaluate $\lvert -26 \rvert - \lvert 15 \rvert$ .

Answer choices for question 16 A $41$ B $11$ C $-11$ D $-41$
17

Why does $(-3) \times (-5)$ have to be positive? Which reason settles it?

Answer choices for question 17 A Because $(-3) \times 0 = 0$ , and spreading the $-3$ across $5 + (-5)$ turns that same zero into $-15$ plus $(-3) \times (-5)$ , so the second product has to be $15$ . B Because two minus signs cancel each other out, in the way that $-(-5) = 5$ . C Because multiplying is repeated addition, so $-3$ groups of $-5$ reverse the direction twice. D Because the sign chart for multiplication lists a negative times a negative as positive.
18

Evaluate $-9 + (-4) \times (-5)$ .

Answer choices for question 18 A $65$ B $-29$ C $-11$ D $11$
19

A cave survey records heights from the entrance as signed numbers, with a point above the entrance recorded as positive and a point below it as negative. Marker A sits at $-38$ meters and marker B at $17$ meters. What signed number records the change in height going from marker A to marker B?

Answer choices for question 19 A $-55$ B $55$ C $-21$ D $21$
20

Evaluate $-\lvert (-7) \times 4 \rvert$ .

Answer choices for question 20 A $28$ B $-3$ C $-28$ D $3$

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Where you stand

Compare and order integers by position on the number line. Review: Negative Numbers and the Number Line
Identify integers and opposites, and handle zero correctly (it has no sign and is its own opposite). Review: Negative Numbers and the Number Line
Add and subtract integers, including subtracting a negative. Review: Adding and Subtracting Integers
Multiply and divide integers and determine the sign of the result. Review: Multiplying and Dividing Integers
Why division by zero is undefined, and what a zero factor does to a product. Review: Multiplying and Dividing Integers
Evaluate absolute value as distance from zero, with the bars acting as a grouping symbol. Review: Absolute Value
Distinguish order from distance from zero, and find the distance between two integers. Review: Absolute Value

Core practice

10 problems from across the chapter. 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 Four labeled marks

On a number line, mark A is halfway between $-2$ and $-1$ . Mark B is the reflection of $0$ across zero. Mark C is nineteen one-unit steps left of zero, and mark D is twenty-four one-unit steps right of zero. Which marks sit at integers? Give the integer at each.
1. Hint 1
  Integers are the whole numbers and their negatives, including zero.
2. Hint 2
  Check whether each position is reached by a whole number of one-unit steps from zero; reflection across zero preserves distance from zero.
Answer

B: $0$ ; C: $-19$ ; D: $24$ . A does not sit at an integer.

Full solution

A lies strictly between two neighboring integer ticks, so it does not sit at an integer.
B is at the point where the line is folded, so reflection keeps it at zero.
$-0=0$

C is at $-19$ , since it is nineteen unit steps to the left.
D is at $24$ , since it is twenty-four unit steps to the right.
Those two numbers and $0$ are integers.

As a check, B, C and D can all be reached from zero in a whole number of unit steps; A cannot.

Answer

B: $0$ ; C: $-19$ ; D: $24$ . A does not sit at an integer.

Key idea

Whole numbers and their negatives are integers, and reflecting zero leaves it at zero.
Problem 1: Mark complete
Problem 2 A box to fill

Find the integer that makes $(-4)\times\square\times(-13)=-156$ true.
1. Hint 1
  The two known factors can be combined first, since the order of multiplication does not change a product.
2. Hint 2
  Multiply $-4$ by $-13$ , then ask which number times that result gives $-156$ .
Answer

$-3$ .

Full solution

Multiplication can be done in any order, so combine the known factors first.
Two negative factors give a positive product.
$(-4)\times(-13)=52$

The equation becomes $52\times\square=-156$ , so the missing factor is a quotient.
The product is negative and $52$ is positive, so the missing factor is negative.
$(-156)\div52=-3$

Check in the original order.
$\begin{aligned}(-4)\times(-3)&=12\\12\times(-13)&=-156\end{aligned}$

Answer

$-3$ .

Key idea

A missing factor is a quotient, and its sign is set by the count of negative factors: odd for a negative product, even for a positive one.
Problem 2: Mark complete
Problem 3 One written expression

Evaluate $\lvert-\lvert-22\rvert\rvert$ .
1. Hint 1
  Each pair of bars measures the distance from zero of the complete value it encloses.
2. Hint 2
  Work outward from the innermost pair, applying the intervening minus sign before the outer pair.
Answer

$22$ .

Full solution

The inner bars enclose $-22$ , so they return its distance from zero.
$\lvert-22\rvert=22$

The minus sign between the pairs takes the opposite of that result.
$-(22)=-22$

The outer bars then measure the resulting number.
$\lvert-22\rvert=22$

As a check, bars and a minus sign change only the sign of $22$ , never its size, and the outer bars never return a negative number, so the result must be $22$ .

Answer

$22$ .

Key idea

With nested bars, complete each inner operation before measuring the value enclosed by the next pair.
Problem 3: Mark complete
Problem 4 Two clock adjustments

Two clocks show offsets from the correct time at the same instant. A negative offset means the clock is behind. Clock A has offset $-17$ seconds and clock B has offset $-6$ seconds.

Clock A is advanced by 9 seconds, and clock B is set back by 4 seconds. Give each new offset and identify which clock then shows the later time.
1. Hint 1
  Advancing a clock increases its offset, while setting it back decreases its offset.
2. Hint 2
  Calculate each clock's new signed offset before comparing the two readings.
3. Hint 3
  The greater offset corresponds to the later displayed time, even if both offsets are negative.
Answer

A: $-8$ seconds. B: $-10$ seconds. Clock A shows the later time.

Full solution

Advance A by adding $9$ to its offset.
$-17+9=-8$
Clock A is now 8 seconds behind.

Set B back by subtracting $4$ from its offset.
$-6-4=-10$
Clock B is now 10 seconds behind.

Compare the new offsets.
$-8>-10$
Clock A has the greater offset, so it displays the later time.

Check by undoing the adjustments.
$\begin{aligned}-8-9&=-17\\-10+4&=-6\end{aligned}$
The original offsets are recovered.

Answer

A: $-8$ seconds. B: $-10$ seconds. Clock A shows the later time.

Key idea

Update each signed quantity before comparing the resulting positions.
Problem 4: Mark complete
Problem 5 A mixed entry

Evaluate $\frac{14-(-10)}{-6}+(-3)\times2$ .
1. Hint 1
  The fraction bar groups the numerator, while the separate product must also be finished before the final addition.
2. Hint 2
  Subtracting the negative number in the numerator increases its value.
3. Hint 3
  After the numerator is complete, determine the signs of the quotient and the separate product.
Answer

$-10$ .

Full solution

First finish the grouped numerator.
$14-(-10)=24$

The denominator is $-6$ , which is nonzero.
The quotient has opposite signs in its numerator and denominator, so it is negative.
$24\div(-6)=-4$

The separate product has one negative factor.
$(-3)\times2=-6$
Add the two completed results.
$-4+(-6)=-10$

Check the quotient by multiplication:
$(-4)\times(-6)=24$
The two negative contributions have sizes $4$ and $6$ , which add to $10$ , consistent with the final sign and size.

Answer

$-10$ .

Key idea

Finish grouped calculations, products and quotients before combining their signed results.
Problem 5: Mark complete
Problem 6 A two-part expression

Evaluate $\lvert-12+5\rvert-\lvert3-11\rvert$ .
1. Hint 1
  Each pair of bars acts on its own enclosed calculation, and the subtraction between the pairs happens afterward.
2. Hint 2
  Finish the sum in the first pair and the subtraction in the second pair before measuring either result.
Answer

$-1$ .

Full solution

Complete the first enclosed sum, then measure its result.
$\begin{aligned}-12+5&=-7\\\lvert-7\rvert&=7\end{aligned}$

Complete the second enclosed difference and measure it.
$\begin{aligned}3-11&=-8\\\lvert-8\rvert&=8\end{aligned}$

The subtraction outside both pairs remains to be done.
$7-8=-1$

Check by adding back the second distance.
$-1+8=7$
Each distance is zero or positive, but subtracting the larger distance from the smaller gives a negative result.

Answer

$-1$ .

Key idea

Bars return distances that are zero or positive, while an operation outside the bars can produce a negative final result.
Problem 6: Mark complete
Problem 7 Two display entries

A programmer replaces $\lvert(-8)\times(-11)\rvert$ with $\lvert-8\rvert\times(-11)$ , claiming that the value stays the same. Find both values and assess the claim.
1. Hint 1
  Identify the complete quantity enclosed by the bars in each expression.
2. Hint 2
  In the first expression, multiply before applying the bars; in the second, apply the bars to their enclosed number before multiplying.
Answer

Original: $88$ . Replacement: $-88$ . The claim is false.

Full solution

The first pair of bars encloses the entire product.
Two negative factors give a positive product.
$(-8)\times(-11)=88$
Its distance from zero is unchanged.
$\lvert88\rvert=88$

In the replacement, the bars enclose just the first factor.
$\lvert-8\rvert=8$
The remaining multiplication has one negative factor.
$8\times(-11)=-88$

The two results have opposite signs, so the replacement changes the value.
Check their sizes: both come from $8\times11$ , which is $88$ , while the position of the bars determines which signs remain.

Answer

Original: $88$ . Replacement: $-88$ . The claim is false.

Key idea

The extent of a pair of bars determines whether a sign is measured away or participates in a later multiplication.
Problem 7: Mark complete
Problem 8 Two moving markers

Markers A and B begin at integer positions on a number line. Reflecting A's starting position across zero gives B's starting position. A moves 7 one-unit steps to the right, while B moves 4 one-unit steps to the left.

Can the sum of their final labels be found without knowing where the markers start? If so, give the sum and explain, including the case when both markers start at zero.
1. Hint 1
  Use the reflection to relate the two starting labels.
2. Hint 2
  Separate each starting label from the change made to it, then consider the total of the starting labels.
3. Hint 3
  Combine the two changes after accounting for the starting pair.
Answer

Yes; the sum is $3$ for every starting pair, including when both markers start at zero.

Full solution

The reflection means the starting labels are opposites.
Write them as $a$ and $-a$ , where $a$ is an integer.
They sum to zero.
$a+(-a)=0$

The first movement adds $7$ and the second adds $-4$ .
Reordering and grouping the additions brings the two starting labels together and the two changes together.
The starting labels contribute zero, and the changes contribute
$7+(-4)=3$
The final total is therefore
$0+3=3$
The letter $a$ no longer appears, so the sum is the same for every starting pair.

If both markers start at zero, their final labels are $7$ and $-4$ , with sum $3$ .
For another check, start at $-10$ and $10$ : the final labels are $-3$ and $6$ , which also sum to $3$ .

Answer

Yes; the sum is $3$ for every starting pair, including when both markers start at zero.

Key idea

When the starting labels are opposites, the sum of the final labels depends only on the two changes.
Problem 8: Mark complete
Problem 9 A trail report

A straight trail has signed distance markers in kilometers, with positive positions east of zero and negative positions west of zero. A cabin is at $-25$ kilometers and a mast is at $-8$ kilometers.

A report says, "The mast is closer to zero, so it is west of the cabin." Assess the report and give the distance between the cabin and the mast.
1. Hint 1
  A position determines east or west, while a distance records the size of a gap.
2. Hint 2
  Place both negative labels on the western side of zero and decide which is farther right.
3. Hint 3
  Count the gap between the two positions rather than adding their distances from zero.
Answer

The report's conclusion is wrong: the mast is closer to zero, but it is east of the cabin. They are 17 kilometers apart.

Full solution

The cabin is 25 kilometers west of zero, while the mast is 8 kilometers west.
The mast is farther right on the line.
$-8>-25$
Thus the mast is east of the cabin, although both are west of zero.

The segment from the cabin to zero consists of the cabin-to-mast gap followed by 8 kilometers from the mast to zero.
Subtract those 8 kilometers from the full 25 kilometers.
$25-8=17$
They are 17 kilometers apart.

Check by joining the two segments again.
$17+8=25$
Being closer to zero on the negative side places the mast farther east, so the report has the direction wrong.

Answer

The report's conclusion is wrong: the mast is closer to zero, but it is east of the cabin. They are 17 kilometers apart.

Key idea

For two negative positions, the one closer to zero is the greater and lies farther east, and the gap between them is the difference of their distances from zero.
Problem 9: Mark complete
Problem 10 Two spreadsheet entries

Two spreadsheet entries are written as $\frac{-8+13}{-5-(-5)}$ and $\frac{-8+8}{-5-(-5)}$ . A reviewer says that each entry has value $0$ . Is the reviewer right about either entry? Explain.
1. Hint 1
  A quotient must specify one number that recovers the numerator when multiplied by the denominator.
2. Hint 2
  Evaluate both numerators and the common denominator before judging the claim.
3. Hint 3
  Consider how many numbers could satisfy each corresponding multiplication.
Answer

The reviewer is wrong about both entries: neither has a value, and each is undefined.

Full solution

The common denominator is found by subtracting a negative.
$-5-(-5)=0$
The numerators are
$\begin{aligned}-8+13&=5\\-8+8&=0\end{aligned}$

The first expression would require a number $q$ satisfying
$q\times0=5$
Every product with a zero factor equals zero, so no number satisfies this condition.
The first expression is undefined.

The second expression would require
$q\times0=0$
Every number satisfies that condition.
For example, both $q=0$ and $q=2$ work, so the condition does not select a unique quotient.
The second expression is also undefined.

Neither expression permits division, since its denominator is zero.
In the first case there is no matching factor, and in the second there is no single matching factor; reporting zero is invalid in both cases.

Answer

The reviewer is wrong about both entries: neither has a value, and each is undefined.

Key idea

A denominator that works out to zero leaves no single quotient, so the expression has no value whatever its numerator.
Problem 10: Mark complete

Star problems Ten optional challenges to stretch your reasoning Advanced. This problem set goes beyond core Pre-Algebra. You can skip it. → Continue to Chapter 3: Factors and Multiples →

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