Integers, Absolute Value, and Number Lines Practice Questions — ACT

Question 1Question →

On a standard number line, point $A$ is located at $-7$ and point $B$ is located at $5$ . What is the distance between point $A$ and point $B$ ?

Show answer & explanation

Answer: 12

Answer

The distance between the two points is 12.

The distance between two points on a number line is the absolute value of their difference. Calculating

|-7 - 5|

yields

|-12|

, which is

12

. Alternatively, subtracting the lesser coordinate from the greater coordinate gives

5 - (-7) = 5 + 7 = 12

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the number line.

The coordinates are

-7

for point

A

and

5

for point

B

.

These coordinates represent the positions from which we need to find the distance.

2

Apply the absolute value formula for the distance between two points,

|a - b|

.

The expression is

|-7 - 5|

.

Distance on a number line is always non-negative and is defined by the absolute value of the difference between the two coordinates.

3

Perform the subtraction and find the absolute value.

|-12| = 12

.

Subtracting

5

from

-7

gives

-12

, and the absolute value of

-12

is

12

.

Key Concept

The distance between two points

a

and

b

on a number line is given by

|a - b|

.

Estimated Time:45s

Question 2Question →

What is the value of the expression $-|-15 + 7|$ ?

-22

-15

-8

8

22

Show answer & explanation

Answer: $-8$

Answer

-8

To find the value of

-|-15 + 7|

, we first evaluate the expression inside the absolute value bars:

-15 + 7 = -8

. Next, we take the absolute value of that result:

|-8| = 8

. Finally, we apply the negative sign that is outside the absolute value bars, which gives

-8

.

Step-by-Step Solution

1

Evaluate the expression inside the absolute value bars.

-15 + 7 = -8

Before applying the absolute value, the terms inside the vertical bars must be simplified according to the order of operations.

2

Find the absolute value of the result from Step 1.

|-8| = 8

The absolute value of a number represents its distance from zero on a number line, which is always non-negative.

3

Apply the negative sign that is outside the absolute value bars.

- (8) = -8

The negative sign outside acts as multiplication by

-1

after the absolute value has been evaluated.

Key Concept

Evaluating expressions involving absolute values and negative numbers

Question 3Question →

What is the value of the expression $|-12| - |7 - 15|$ ?

Show answer & explanation

Answer: 4

Answer

The value of the expression is 4.

The absolute value of a number represents its distance from zero on a number line, so

|-12| = 12

. Simplifying inside the second term gives

7 - 15 = -8

, and the absolute value

|-8| = 8

. Subtracting these two results yields

12 - 8 = 4

.

Step-by-Step Solution

1

Evaluate the first term

|-12| = 12

The absolute value of a number is its distance from zero on the number line, which is always non-negative.

2

Simplify the expression inside the second absolute value

7 - 15 = -8

Subtracting 15 from 7 yields a negative difference of -8.

3

Evaluate the second absolute value term

|-8| = 8

The absolute value of -8 is 8.

4

Subtract the two evaluated terms

12 - 8 = 4

Subtracting 8 from 12 gives the final simplified value of 4.

Key Concept

Evaluating expressions with absolute values requires simplifying the terms inside the absolute value grouping symbols first, taking the absolute value of the result, and then performing the subtraction.

Question 4Question →

The temperature at the top of a mountain is $-14^\circ\text{C}$ , and the temperature at the base of the mountain is $8^\circ\text{C}$ . What is the absolute difference, in degrees Celsius, between these two temperatures?

-22

-6

6

14

22

Show answer & explanation

Answer: 22

Answer

22

The correct answer is 22 because the absolute difference between

8

and

-14

is

|8 - (-14)| = |8 + 14| = |22| = 22

. This represents the total distance of 22 units between the two temperatures on a number line.

Step-by-Step Solution

1

Translate the problem into a mathematical expression using absolute value for the difference between the two temperatures.

The absolute difference is given by

|8 - (-14)|

or

|-14 - 8|

.

The absolute difference between two numbers

a

and

b

on a number line is defined as

|a - b|

.

2

Simplify the subtraction inside the absolute value.

8 - (-14) = 8 + 14 = 22

(or

-14 - 8 = -22

).

Subtracting a negative number is equivalent to adding its positive counterpart.

3

Evaluate the absolute value.

|22| = 22

(or

|-22| = 22

).

The absolute value of a number represents its non-negative distance from zero.

Key Concept

Integers, Absolute Value, and Number Lines

Estimated Time:45s

Question 5Question →

Four distinct integers, $p$ , $q$ , $r$ , and $s$ , are represented on a standard number line. The distance between $p$ and $q$ is 3, the distance between $q$ and $r$ is 4, and the distance between $r$ and $s$ is 5. What is the minimum possible distance between $p$ and $s$ ?

Show answer & explanation

Answer: 2

Answer

The minimum possible distance between

p

and

s

is 2.

By setting

p = 0

as a reference point, the coordinate for

q

must be either

3

or

-3

. Assuming

q = 3

by symmetry,

r

must be either

-1

or

7

because it is at a distance of 4 from

q

. From

r = 7

, a distance of 5 leads to

s

being at

2

or

12

. From

r = -1

, a distance of 5 leads to

s

being at

4

or

-6

. All of these positions result in four distinct integers. The distances between

p

and

s

are the absolute values of their coordinates, which are

12

,

2

,

6

, and

4

. The minimum of these distances is

2

.

Step-by-Step Solution

1

Establish a coordinate system on the number line.

Let

p

be at position 0. Since the distance between

p

and

q

is 3,

q

is at either

3

or

-3

. By symmetry, we assume

q = 3

.

Setting one point at the origin simplifies the relative distance calculations for all other points.

2

Find the possible coordinates of

r

.

Since the distance between

q

and

r

is 4,

r

is at

3 - 4 = -1

or

3 + 4 = 7

.

The absolute value equation

|q - r| = 4

has two solutions for

r

given

q = 3

.

3

Find the possible coordinates of

s

for each case of

r

.

If

r = 7

, then

s

can be at

7 - 5 = 2

or

7 + 5 = 12

. If

r = -1

, then

s

can be at

-1 - 5 = -6

or

-1 + 5 = 4

. All generated sets contain distinct values, satisfying the requirement.

The absolute value equation

|r - s| = 5

has two solutions for

s

for each candidate coordinate of

r

.

4

Determine the minimum distance between

p

and

s

.

The possible values for the distance

|p - s|

are

|0 - 2| = 2

,

|0 - 12| = 12

,

|0 - (-6)| = 6

, and

|0 - 4| = 4

. The minimum value is 2.

Comparing all possible valid configurations ensures we find the true minimum distance.

Key Concept

Representing distances between points on a number line using absolute values and resolving configurations for distinct integers.

Question 6Question →

If $x = -6$ , what is the value of the expression $12 - |x - 3|$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is 3.

Substituting

x = -6

into the expression

12 - |x - 3|

gives

12 - |-6 - 3|

. Simplifying the subtraction inside the absolute value gives

12 - |-9|

. Since the absolute value of

-9

is

9

, the expression simplifies to

12 - 9

, which equals

3

.

Step-by-Step Solution

1

Substitute

x = -6

into the expression.

12 - |-6 - 3|

Substitute the given value of the variable into the algebraic expression.

2

Simplify the expression inside the absolute value symbol.

12 - |-9|

Subtracting 3 from

-6

results in

-9

.

3

Calculate the absolute value.

12 - 9

The absolute value represents the distance from zero on the number line, so

|-9| = 9

.

4

Subtract the numbers to find the final value.

3

12 - 9 = 3

.

Key Concept

Evaluating algebraic expressions involving integers and absolute values.

Question 7Question →

Let $x$ , $y$ , and $z$ be integers such that $|x| < |y| < |z|$ , $x + y + z = -3$ , and $xyz > 0$ . What is the smallest possible value of $|x - y| + |y - z| + |z - x|$ ?

6

8

10

12

14

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. To satisfy the given conditions, the product

xyz > 0

means either all three numbers are positive, or exactly two are negative. Since they sum to

-3

, they cannot all be positive. Letting two be negative and one positive, we find that the integer set

\{-1, 2, -4\}

satisfies

|-1| < |2| < |-4|

(or

1 < 2 < 4

), has a sum of

-3

, and a product of

8 > 0

. The sum of the absolute differences is

|-1 - 2| + |2 - (-4)| + |-4 - (-1)| = 3 + 6 + 3 = 12

, which is the minimum possible value.

Step-by-Step Solution

1

Analyze the sign constraints from the product condition

xyz > 0

.

Since the product of

x

,

y

, and

z

is positive, either all three integers are positive, or exactly two are negative and one is positive.

The product of three numbers is positive if there are zero or two negative factors.

2

Evaluate the case where all three integers are positive.

If all three are positive, then

x, y, z \geq 1

. This implies their sum

x+y+z \geq 3

, which contradicts the constraint

x+y+z = -3

. Thus, this case is impossible.

Positive integers cannot sum to a negative number.

3

Analyze the case where two integers are negative and one is positive, and express them in terms of their absolute values.

Let the negative integers be

-u

and

-v

(with

u > v > 0

) and the positive integer be

p > 0

. The sum constraint becomes

-u - v + p = -3

, which simplifies to

p = u + v - 3

. The absolute values are

u

,

v

, and

p

, which must be three distinct positive integers satisfying

|x| < |y| < |z|

.

This translates the constraints into positive integer variables representing the absolute values.

4

Determine the values of

u, v, p

that minimize the maximum difference between any two of the three integers

x, y, z

.

For

v = 1

: if

u = 4

, then

p = 2

. The absolute values

\{1, 2, 4\}

are distinct. The corresponding integers are

x = -1

,

y = 2

, and

z = -4

. Their sum is

-3

and product is

8 > 0

. The difference between the maximum and minimum values is

2 - (-4) = 6

. For

v = 2

: if

u = 4

, then

p = 3

. The absolute values

\{2, 3, 4\}

are distinct, giving the integers

-4, -2, 3

, with a maximum difference of

3 - (-4) = 7

. For higher values, the difference only increases.

The expression

|x - y| + |y - z| + |z - x|

is equal to twice the difference between the maximum and minimum of the three integers.

5

Calculate the minimum value of the expression.

Using the optimal set

x = -1, y = 2, z = -4

, the value of the expression is

|-1 - 2| + |2 - (-4)| + |-4 - (-1)| = 3 + 6 + 3 = 12

.

This is the smallest possible sum of the absolute differences.

Key Concept

Using absolute values and sign analysis to solve system of constraints on integers and distances on a number line.

Question 8Question →

What integer represents the midpoint between $-9$ and $7$ on a standard number line?

8

1

-1

-2

16

Show answer & explanation

Answer: $-1$

Answer

The midpoint coordinate is

-1

.

The midpoint of two numbers on a number line is their mathematical average. Adding the coordinates

-9

and

7

gives a sum of

-2

. Dividing this sum by

2

yields the midpoint coordinate of

-1

.

Step-by-Step Solution

1

Identify the coordinates of the two endpoints on the number line.

The endpoints are

-9

and

7

.

Before calculating the midpoint, we must identify the values of the two points given in the problem.

2

Find the average of the two coordinates using the midpoint formula

\frac{a + b}{2}

.

\frac{-9 + 7}{2} = \frac{-2}{2} = -1

.

The midpoint of any two coordinates on a number line is their arithmetic mean.

Key Concept

Finding the midpoint between two integers on a number line

Estimated Time:45s

Question 9Question →

Let $x$ and $y$ be integers such that $|x| \leq 4$ and $|y| \leq 4$ . How many distinct pairs of integers $(x, y)$ satisfy the inequality $||x| - |y|| \geq 2$ ?

Show answer & explanation

Answer: 36

Answer

There are 36 distinct pairs of integers

(x, y)

that satisfy the inequality.

By analyzing the possible values for

|x|

and

|y|

within the set

\{0, 1, 2, 3, 4\}

, we identify the 12 pairs that satisfy the inequality

||x| - |y|| \geq 2

. When mapping these absolute values back to the actual integer coordinates, we account for the single option when a coordinate is 0 and the two positive/negative options when a coordinate is non-zero. Summing these possibilities yields exactly 36 distinct pairs.

Step-by-Step Solution

1

Determine the range of absolute values for the integers.

The possible values for

|x|

and

|y|

are

\{0, 1, 2, 3, 4\}

.

Since

x

and

y

are integers satisfying

-4 \leq x, y \leq 4

, their absolute values must be non-negative integers up to 4.

2

Find the pairs of absolute values

(|x|, |y|)

that satisfy the inequality.

The valid pairs are:

(0, 2), (0, 3), (0, 4), (1, 3), (1, 4), (2, 0), (2, 4), (3, 0), (3, 1), (4, 0), (4, 1), (4, 2)

.

We test all combinations of

u, v \in \{0, 1, 2, 3, 4\}

such that

|u - v| \geq 2

.

3

Calculate the number of integer coordinate pairs

(x, y)

corresponding to each absolute value pair.

Pairs with one zero value yield 2 integer solutions (e.g.,

u=0, v=2 \Rightarrow (0, 2), (0, -2)

). Pairs with both non-zero values yield 4 integer solutions (e.g.,

u=1, v=3 \Rightarrow (1, 3), (1, -3), (-1, 3), (-1, -3)

).

An absolute value of 0 corresponds to only 1 integer (

0

), whereas any positive absolute value

k

corresponds to 2 integers (

k

and

-k

).

4

Sum the number of integer pairs for all valid cases.

Total pairs =

(3 \times 2) + (3 \times 2) + (6 \times 4) = 6 + 6 + 24 = 36

.

There are 3 pairs with

u=0

(

6

solutions), 3 pairs with

v=0

(

6

solutions), and 6 pairs with both

u, v > 0

(

24

solutions).

Key Concept

Solving nested absolute value inequalities with integer constraints and counting solution pairs systematically.

Question 10Question →

Point $P$ is located at $-14$ on a standard number line. Point $Q$ is located $8$ units from point $P$ in the positive direction. Point $R$ is located $11$ units from point $Q$ in the negative direction. What integer represents the location of point $R$ ?

Show answer & explanation

Answer: -17

Answer

The position of point

R

is represented by the integer

-17

.

To find the location of point

R

, we first determine the location of point

Q

by moving

8

units in the positive direction (right) from point

P

(at

-14

), which gives

-14 + 8 = -6

. Next, we find the location of point

R

by moving

11

units in the negative direction (left) from point

Q

(at

-6

), which gives

-6 - 11 = -17

.

Step-by-Step Solution

1

Calculate the position of point

Q

.

-6

Since point

Q

is

8

units from point

P

(which is at

-14

) in the positive direction, we add

8

to

-14

:

-14 + 8 = -6

.

2

Calculate the position of point

R

.

-17

Since point

R

is

11

units from point

Q

(which is at

-6

) in the negative direction, we subtract

11

from

-6

:

-6 - 11 = -17

.

Key Concept

Adding and subtracting integers on a number line to find positions relative to a starting point.

Estimated Time:45s

Question 11Question →

On a standard number line, the distance between two points, $A$ and $B$ , is $12$ units. If the coordinate of point $A$ is $-18$ , which of the following is a possible coordinate for point $B$ ?

-30

-12

-8

6

30

Show answer & explanation

Answer: $-30$

Answer

-30

The correct answer is

-30

because the distance between two points on a number line is found by taking the absolute value of the difference between their coordinates. Setting up the relation

|b - (-18)| = 12

simplifies to

|b + 18| = 12

. Solving this absolute value equation yields two possible locations:

b + 18 = 12

(which gives

b = -6

) and

b + 18 = -12

(which gives

b = -30

). Among the choices,

-30

is the only possible coordinate listed.

Step-by-Step Solution

1

Set up the absolute value equation representing the distance between point A and point B on the number line.

Let

b

be the coordinate of point

B

. The distance between

A

and

B

is given by

|-18 - b| = 12

.

On a standard number line, the distance between any two coordinates

x

and

y

is given by the absolute value of their difference,

|x - y|

.

2

Solve the absolute value equation for the two possible cases.

Case 1:

-18 - b = 12 \Rightarrow -b = 30 \Rightarrow b = -30

. Case 2:

-18 - b = -12 \Rightarrow -b = 6 \Rightarrow b = -6

.

An equation of the form

|x| = d

(where

d \geq 0

) splits into two distinct equations:

x = d

and

x = -d

.

3

Compare the possible solutions with the given options.

The value

-30

is one of the possible coordinates and is listed in the options.

To identify which of the two mathematically correct coordinates (-30 or -6) is provided as an answer choice.

Key Concept

The distance between two points on a number line with coordinates

x

and

y

is represented by the absolute value of their difference,

|x - y|

.

Estimated Time:45s

Question 12Question →

Three distinct integers, $a$ , $b$ , and $c$ , lie on a standard number line such that $a < b < c$ . The distance between $a$ and $b$ is $3$ times the distance between $b$ and $c$ . If $|a| = 15$ , $|c| = 7$ , and $b < 0$ , what is the value of $b$ ?

Show answer & explanation

Answer: -9

Answer

The value of

b

is

-9

.

The correct answer is

-9

. By interpreting

|a| = 15

and

|c| = 7

with the constraint

a < b < c

, we find

a = -15

. Testing the possible values for

c

, when

c = -7

, we set up the distance equation

b - (-15) = 3(-7 - b)

, which simplifies to

b + 15 = -21 - 3b

, giving

4b = -36

and

b = -9

. This satisfies all constraints, including

b

being a negative integer.

Step-by-Step Solution

1

Find the possible coordinates for

a

and

c

based on their absolute values.

a \in \{-15, 15\}

and

c \in \{-7, 7\}

.

The absolute value of a number represents its distance from zero, so

|x| = d \implies x = \pm d

.

2

Use the ordering condition

a < b < c

to eliminate invalid combinations.

a = -15

and

c \in \{-7, 7\}

.

Since

a

must be less than

c

,

a

cannot be

15

because both possible values of

c

(

-7

and

7

) are less than

15

.

3

Set up an equation representing the distance relationship on the number line.

b + 15 = 3(c - b) \implies 4b = 3c - 15

.

For points on a number line ordered

a < b < c

, the distance between

a

and

b

is

b - a

, and the distance between

b

and

c

is

c - b

.

4

Substitute each possible value of

c

and solve for

b

to find the one that yields a negative integer.

For

c = -7

,

b = -9

.

If

c = 7

,

b = 1.5

, which is not an integer. If

c = -7

,

b = -9

, which is a negative integer, satisfying all given conditions.

Key Concept

Using absolute values and relative order to determine integer positions and distances on a number line.

Question 13Question →

On a standard number line, the coordinates of three distinct points $A$ , $B$ , and $C$ are represented by the integers $a$ , $b$ , and $c$ , respectively, such that $a < b < c$ . The distance between each point and the origin is represented by its absolute value, and these distances satisfy the inequality $|a| < |b| < |c|$ . If the product of the three coordinates is negative ( $abc < 0$ ) and the sum of their absolute values is $20$ , what is the maximum possible value of the coordinate $b$ ?

6

8

9

10

18

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. By analyzing the signs of the coordinates, we determine that only

a

is negative, so

a < 0 < b < c

. Since

a

is a negative integer, the minimum value of

|a|

is 1. Under the constraint

|a| < |b| < |c|

and

|a| + |b| + |c| = 20

, we let

x = -a

,

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we get

20 = x + y + z > 1 + 2y

, which implies

2y < 19

, or

y < 9.5

. The maximum integer value for

y

(which is

b

) is 9, achieved when the coordinates are -1, 9, and 10.

Step-by-Step Solution

1

Determine the signs of the coordinates based on the given constraints.

The sign configuration must be

a < 0 < b < c

.

Since the product

abc < 0

, either all three coordinates are negative or exactly one is negative. If all three were negative (

a < b < c < 0

), their absolute values would satisfy

|a| > |b| > |c|

, which contradicts

|a| < |b| < |c|

. Thus, exactly one coordinate (

a

) is negative, and the other two (

b

and

c

) are positive.

2

Translate the absolute values and sum constraint into algebraic terms.

-a + b + c = 20

, where

a < 0

and

b, c > 0

.

For a negative number

a

, the absolute value

|a| = -a

. For positive numbers

b

and

c

,

|b| = b

and

|c| = c

. Therefore, the sum

|a| + |b| + |c| = 20

simplifies to

-a + b + c = 20

.

3

Set up inequalities to bound the coordinate of the middle point.

1 \le -a < b < c

.

Since

a

is a non-zero negative integer, the smallest possible value for its absolute value

|a| = -a

is

1

. The condition

|a| < |b| < |c|

then becomes

1 \le -a < b < c

.

4

Use the constraints to find the maximum value of

b

.

The maximum possible value of

b

is

9

.

Let

x = -a

,

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we have

20 = x + y + z > x + 2y \ge 1 + 2y

. This simplifies to

19 > 2y

, or

y < 9.5

. Since

y

must be an integer, the maximum possible value for

y

(which is

b

) is

9

. We can verify this maximum by setting

a = -1

,

b = 9

, and

c = 10

, which perfectly satisfies all conditions.

Key Concept

Analyzing coordinate signs and absolute value inequalities on a number line to perform optimization under integer constraints.

Question 14Question →

Let $S$ be the set of all integers $n$ that satisfy the inequality $|n - 3| + |n + 5| \leq 12$ . What is the sum of the absolute values of all integers in set $S$ ?

43

-13

12

28

37

Show answer & explanation

Answer: 43

Answer

43

The correct answer is 43. Solving the inequality

|n - 3| + |n + 5| \leq 12

by casework on the number line yields the solution set of integers

S = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

. Summing the absolute values of these elements gives

7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 43

.

Step-by-Step Solution

1

Divide the number line into intervals to remove the absolute value bars based on critical points

n = 3

and

n = -5

.

Three cases to analyze: Case 1 (

n \geq 3

), Case 2 (

-5 \leq n < 3

), and Case 3 (

n < -5

).

Evaluating absolute values requires knowing the signs of the expressions inside them.

2

Solve the inequality

|n - 3| + |n + 5| \leq 12

for each case.

For Case 1 (

n \geq 3

):

(n - 3) + (n + 5) \leq 12 \Rightarrow 2n + 2 \leq 12 \Rightarrow n \leq 5

, yielding integers

\{3, 4, 5\}

. For Case 2 (

-5 \leq n < 3

):

-(n - 3) + (n + 5) \leq 12 \Rightarrow 8 \leq 12

(always true), yielding integers

\{-5, -4, -3, -2, -1, 0, 1, 2\}

. For Case 3 (

n < -5

):

-(n - 3) - (n + 5) \leq 12 \Rightarrow -2n - 2 \leq 12 \Rightarrow -2n \leq 14 \Rightarrow n \geq -7

, yielding integers

\{-7, -6\}

.

Determining the integer values of

n

that satisfy the inequality in each segment of the number line.

3

Combine the intervals to construct the complete set

S

.

S = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

.

The union of all valid cases gives the entire set of solutions.

4

Compute the sum of the absolute values of the integers in

S

.

|-7| + |-6| + |-5| + |-4| + |-3| + |-2| + |-1| + |0| + |1| + |2| + |3| + |4| + |5| = 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 43

.

Adding the absolute values of each element in the solution set.

Key Concept

Solving multi-interval absolute value inequalities with integers and finding absolute values

Estimated Time:2m 0s

Question 15Question →

On a standard number line, the coordinate of point $P$ is an integer $x$ . If the sum of the distances from $P$ to $-2$ and from $P$ to $4$ is equal to the distance from $P$ to $10$ , what is the sum of all possible values of $x$ ?

Show answer & explanation

Answer: -4

Answer

The sum of all possible values of the integer

x

is

-4

.

The correct answer is

-4

. The distance between any two coordinates

a

and

b

on a number line is defined as

|a - b|

. Translating the problem gives the equation

|x + 2| + |x - 4| = |x - 10|

. Breaking the number line into intervals around the critical points

x = -2

,

x = 4

, and

x = 10

yields two valid integer solutions:

x = -8

and

x = 4

. The sum of these values is

-8 + 4 = -4

.

Step-by-Step Solution

1

Express the distances on the number line using absolute values.

The distance from

P(x)

to

-2

is

|x - (-2)| = |x + 2|

. The distance from

P(x)

to

4

is

|x - 4|

. The distance from

P(x)

to

10

is

|x - 10|

. The equation is

|x + 2| + |x - 4| = |x - 10|

.

The absolute value

|a - b|

represents the distance between points

a

and

b

on a standard number line.

2

Solve the equation by testing the intervals defined by the critical points

x = -2

,

x = 4

, and

x = 10

.

We analyze the four intervals:
- For

x < -2

:

-(x + 2) - (x - 4) = -(x - 10) \implies -2x + 2 = -x + 10 \implies x = -8

. Since

-8 < -2

, this is a valid solution.
- For

-2 \leq x < 4

:

(x + 2) - (x - 4) = -(x - 10) \implies 6 = -x + 10 \implies x = 4

. Since

4

is not in

[-2, 4)

, there is no solution in this interval.
- For

4 \leq x < 10

:

(x + 2) + (x - 4) = -(x - 10) \implies 2x - 2 = -x + 10 \implies 3x = 12 \implies x = 4

. Since

4

is in

[4, 10)

, this is a valid solution.
- For

x \geq 10

:

(x + 2) + (x - 4) = x - 10 \implies 2x - 2 = x - 10 \implies x = -8

. Since

-8 < 10

, there is no solution in this interval.

Absolute value terms change sign at their critical points, requiring case-by-case evaluation.

3

Sum all valid integer solutions.

The valid values for

x

are

-8

and

4

. Their sum is

-8 + 4 = -4

.

The problem asks for the sum of all possible values of

x

.

Key Concept

Representing geometric distances on a number line using absolute value equations and solving them using interval analysis.

Question 16Question →

An explorer records the elevation, in meters relative to sea level, of four research stations: $W$ , $X$ , $Y$ , and $Z$ . Station $W$ is located at an elevation of $-15$ meters. The elevation of Station $X$ is the absolute value of the elevation of Station $W$ . The elevation of Station $Y$ is $8$ meters lower than the elevation of Station $X$ . Station $Z$ is at an elevation such that the distance between the elevations of Station $Y$ and Station $Z$ on a vertical number line is exactly $12$ meters. Which of the following could be the elevation, in meters, of Station $Z$ ?

-35

-11

-5

5

11

Show answer & explanation

Answer: $-5$

Answer

The correct elevation of Station Z could be

-5

meters.

The correct answer is

-5

meters. First, we find the elevation of Station X, which is the absolute value of Station W's elevation:

|-15| = 15

meters. Next, we determine the elevation of Station Y, which is

8

meters lower than Station X:

15 - 8 = 7

meters. Finally, we find the possible elevations for Station Z. The distance between Station Y and Station Z is

12

meters, which can be represented by the equation

|Z - 7| = 12

. This gives two possible elevations:

Z = 7 + 12 = 19

meters or

Z = 7 - 12 = -5

meters. Among the choices, only

-5

is listed.

Step-by-Step Solution

1

Find the elevation of Station X by taking the absolute value of the elevation of Station W.

The elevation of Station X is

|-15| = 15

meters.

The problem states that the elevation of Station X is the absolute value of the elevation of Station W, which is

-15

meters.

2

Calculate the elevation of Station Y by subtracting 8 meters from the elevation of Station X.

The elevation of Station Y is

15 - 8 = 7

meters.

Station Y is 8 meters lower than Station X, which is at 15 meters.

3

Set up the absolute value equation representing the distance of 12 meters between Station Y and Station Z, and solve for the two possible values of Z.

|Z - 7| = 12

, which yields

Z - 7 = 12 \implies Z = 19

or

Z - 7 = -12 \implies Z = -5

.

The distance between two points

a

and

b

on a number line is given by

|a - b|

. Since the distance between Y and Z is 12, we solve the equation to find all possible elevations.

Key Concept

Absolute value represents the distance of a number from zero on a number line, and the distance between two numbers

a

and

b

is given by

|a - b|

.

Estimated Time:2m 0s

Question 17Question →

On a standard number line, points $A$ , $B$ , $C$ , and $D$ have distinct integer coordinates $a$ , $b$ , $c$ , and $d$ , respectively, such that $a < b < c < d$ . The distance between $A$ and $B$ is equal to the distance between $C$ and $D$ . The distance between $B$ and $C$ is $\frac{2}{3}$ of the distance between $A$ and $B$ . If the average (arithmetic mean) of the four coordinates is $0$ and $|a - d| = 24$ , what is the coordinate of $B$ ?

Show answer & explanation

Answer: -3

Answer

The coordinate of

B

is

-3

.

The coordinate of

B

is found by setting the segment lengths to

3k

,

2k

, and

3k

based on the given ratio. Using the absolute value distance

|a - d| = 24

, we find

k = 3

, meaning the segments are

9

,

6

, and

9

. Using the coordinate sum of

0

, we get

4a + 48 = 0 \implies a = -12

. Substituting this back gives the coordinate of

B

as

-3

.

Step-by-Step Solution

1

Define segment lengths using a variable

k

based on the given ratio.

Let the distance

AB = CD = 3k

and

BC = 2k

.

This allows us to write all distances as integer multiples of a single variable since the ratio of

BC

to

AB

is

\frac{2}{3}

.

2

Set up an equation for the total distance from

A

to

D

using the absolute value

|a - d| = 24

.

Since

a < d

,

d - a = |a - d| = 24

. The sum of the segments is

3k + 2k + 3k = 8k

. Solving

8k = 24

yields

k = 3

.

The absolute value of the difference between the outermost points represents the total length of the number line segment containing all four points.

3

Express the coordinates of

b

,

c

, and

d

in terms of

a

using the calculated segment lengths.

b = a + 9

,

c = a + 15

, and

d = a + 24

.

Since the points are in order

a < b < c < d

, we add the segment lengths successively to find the coordinates.

4

Apply the average condition to solve for the coordinate

a

.

The sum of the coordinates is

4 \times 0 = 0

, so

a + (a + 9) + (a + 15) + (a + 24) = 4a + 48 = 0

, giving

a = -12

.

An average of

0

for four numbers means their sum must be

0

.

5

Find the coordinate of

B

by substituting the value of

a

into the expression for

b

.

b = -12 + 9 = -3

.

This yields the specific coordinate requested by the question.

Key Concept

Using absolute value as distance on a number line and expressing relationships between ordered coordinates algebraically.

Question 18Question →

A scientist monitors the temperature in three different chambers, $A$ , $B$ , and $C$ . The temperature in Chamber $B$ is $T^\circ\text{C}$ . The temperature in Chamber $A$ is always $5^\circ\text{C}$ colder than the temperature in Chamber $B$ , and the temperature in Chamber $C$ is always $9^\circ\text{C}$ warmer than the temperature in Chamber $B$ . If $T$ must be an integer, and the sum of the absolute values of the temperatures in all three chambers is minimized, what is the value of $T$ ?

-9

-1

0

2

5

Show answer & explanation

Answer: 0

Answer

The correct answer is

0

.

The correct answer is the value

0

. The sum of the absolute values of the temperatures in all three chambers is represented by the function

S(T) = |T - 5| + |T| + |T + 9|

. Geometrically, this sum represents the total distance from a point

T

to the coordinates

5

,

0

, and

-9

on a number line. For any set of points, the sum of absolute differences is minimized at their median. The median of the set

\{-9, 0, 5\}

is

0

. Evaluating the sum at

T = 0

yields

|0 - 5| + |0| + |0 + 9| = 5 + 0 + 9 = 14

. Any other integer choice yields a larger sum (for example, a value of

1

or

-1

yields a sum of

15

).

Step-by-Step Solution

1

Write expressions for the temperatures of the three chambers in terms of

T

.

The temperature in Chamber

A

is

T - 5

, in Chamber

B

is

T

, and in Chamber

C

is

T + 9

.

To represent the temperature of each chamber relative to Chamber

B

.

2

Write the sum of the absolute values of the three temperatures.

The sum is represented by the expression

S(T) = |T - 5| + |T| + |T + 9|

.

To formulate the function that needs to be minimized.

3

Find the key values that make each absolute value term zero.

The points are

5

,

0

, and

-9

. Ordered on a number line, these points are

-9

,

0

, and

5

.

The sum of absolute differences is geometrically equivalent to the sum of distances on a number line, which is minimized at the median of the points.

4

Evaluate the sum of absolute values at the median value,

T = 0

, and compare it to other values.

For

T = 0

,

S(0) = |-5| + |0| + |9| = 14

. For

T = 1

,

S(1) = |-4| + |1| + |10| = 15

. For

T = -1

,

S(-1) = |-6| + |-1| + |8| = 15

. Thus, the minimum sum occurs at

T = 0

.

To verify that the median yields the minimum sum of absolute values.

Key Concept

Minimizing the sum of absolute values of linear terms by finding the median of their zero-points on a number line.

Question 19Question →

Two integers, $x$ and $y$ , are positioned on a number line. The distance between $x$ and $1$ is twice the distance between $y$ and $2$ . If the distance between $x$ and $y$ is exactly $5$ units, what is the sum of all possible values of $x + y$ ?

Show answer & explanation

Answer: 17

Answer

The sum of all possible values of

x + y

is

17

.

Representing the distances algebraically yields the system

|x - 1| = 2|y - 2|

and

|x - y| = 5

. Since

x

and

y

must be integers, splitting these equations into positive and negative cases yields exactly three valid integer coordinate pairs:

(13, 8)

,

(5, 0)

, and

(-7, -2)

. The sums (

x + y

) for these pairs are

21

,

5

, and

-9

, respectively. Adding these possible sums together gives a final total of

17

.

Step-by-Step Solution

1

Set up the absolute value expressions representing the distances.

|x - 1| = 2|y - 2|

and

|x - y| = 5

Distance on a number line between two points

a

and

b

is mathematically defined as

|a - b|

.

2

Split the distance condition

|x - y| = 5

into two coordinate cases.

x = y + 5

or

x = y - 5

An absolute value equation of the form

|A| = B

splits into

A = B

or

A = -B

.

3

Substitute

x = y + 5

into the first equation and solve for integer values of

y

.

y = 8

(which gives

x = 13

) and

y = 0

(which gives

x = 5

)

This generates the first set of valid integer coordinates satisfying all constraints.

4

Substitute

x = y - 5

into the first equation and solve for integer values of

y

.

y = -2

(which gives

x = -7

); the second algebraic option

y = 10/3

is discarded because it is not an integer

This generates the remaining valid integer coordinates satisfying all constraints.

5

Sum the value of

x + y

for all three valid coordinate pairs.

(13 + 8) + (5 + 0) + (-7 - 2) = 21 + 5 - 9 = 17

The question asks for the sum of all possible values of the expression

x + y

.

Key Concept

Using absolute value to represent distances on a number line and solving systems of absolute value equations under integer constraints.

Estimated Time:2m 30s

Question 20Question →

A chemical mixture is kept in a temperature-controlled chamber where the temperature $T$ , in degrees Celsius, is restricted to the range $-10 \le T \le 10$ . The mixture remains stable as long as the temperature satisfies the inequality:

2 - 3|T - 2| \ge -10

What is the sum of all integer values of $T$ in this range for which the mixture is stable?

19

-14

18

-18

20

Show answer & explanation

Answer: 18

Answer

18

The correct answer is 18. Subtracting 2 from both sides of the inequality

2 - 3|T - 2| \ge -10

gives

-3|T - 2| \ge -12

. Dividing by

-3

and reversing the inequality sign yields

|T - 2| \le 4

, which expands to the compound inequality

-4 \le T - 2 \le 4

. Adding 2 to all parts gives

-2 \le T \le 6

. All integers in this range (

-2, -1, 0, 1, 2, 3, 4, 5, 6

) are within the chamber's allowed range of

[-10, 10]

. Their sum is

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 18

.

Step-by-Step Solution

1

Isolate the absolute value expression by subtracting 2 from both sides.

-3|T - 2| \ge -12

To solve an absolute value inequality, we must first isolate the absolute value term on one side.

2

Divide both sides by

-3

and reverse the inequality sign.

|T - 2| \le 4

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

3

Rewrite the absolute value inequality as a compound inequality.

-4 \le T - 2 \le 4

An inequality of the form

|x| \le a

(where

a \ge 0

) is equivalent to

-a \le x \le a

.

4

Solve for

T

by adding 2 to all three parts of the inequality.

-2 \le T \le 6

Adding 2 isolates the variable

T

to find the range of stable temperatures.

5

Identify all integer values of

T

in the stable range and calculate their sum.

Sum = 18

The integers in the interval

[-2, 6]

are

-2, -1, 0, 1, 2, 3, 4, 5, 6

. Since the chamber is restricted to

[-10, 10]

, all of these values are valid. Summing them:

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 18

.

Key Concept

Solving absolute value inequalities and performing operations with signed integers.

Integers, Absolute Value, and Number Lines

Answer

Step-by-Step Solution

Key Concept

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