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Distance Masters

A comprehensive middle school mathematics unit introducing absolute value through geometric distance, order of operations with magnitude, and dual-branch linear equations.

Distance from Zero

Students uncover the foundational concept of absolute value as distance on a number line, distinguish magnitude from direction, and interpret real-world contexts like elevation and debt.

Expressions and Magnitude

Students evaluate numerical expressions containing absolute value bars as grouping symbols, compare integer magnitudes, and compute physical distances between coordinate points.

Branching Equations

Students solve basic single-variable absolute value equations by setting up dual positive and negative branches, recognizing impossible equations, and modeling real-world tolerance.

Distance Masters

A comprehensive middle school mathematics unit introducing absolute value through geometric distance, order of operations with magnitude, and dual-branch linear equations.

Zero Distance Slides

Lesson 1 • Middle School Math

Distance Masters Series

Zero Distance

Discover why numbers care about how far they travel, not which direction they go.

Distance Concept

Steps from zero

Notation $|x|$

Absolute value bars

Real Scenarios

Elevation & depth

What is Absolute Value?

Distance is always positive or zero — never negative!

Core Concept

"How many steps is a number from 0?"

5 steps away

←

→

-5

-4

-3

-2

-1

0

1

2

3

4

+5

Both -5 and +5 are exactly 5 units away from 0.

$|-5| = 5$ and $|5| = 5$

The Math Notation: $|x|$

Two straight vertical bars hug the number or expression.

Rules

Positive Input

Leaves the number unchanged.

$|8| = 8$

8 units to the right

Negative Input

Strips the negative sign away.

$|-14| = 14$

14 units to the left

Zero Input

Zero has no distance from itself.

$|0| = 0$

Zero steps

Remember: An absolute value output can NEVER be a negative number! Distance cannot be negative.

Real-World Magnitude

When we care about the amount, not the sign.

Applications

Sea Level vs. Submarine

A submarine is at an elevation of -280 meters.

Physical Depth:

$|-280| = 280\text{ meters deep}$

Bank Balance vs. Debt

Maya's account shows a balance of -$45.

Total Debt Owed:

$|-45| = \$45\text{ owed}$

Words like depth, debt, and difference describe absolute values!

The Danger Zone: Signs Outside!

Watch the position of the negative sign closely.

Common Trap

Negative Inside Standard

$|-9| = 9$

The absolute value operation acts first. The distance of -9 from 0 is positive 9.

Negative Outside Opposite of...

$-|-9| = -9$

First evaluate $|-9| = 9$. Then apply the negative sign waiting outside: result is -9!

Treat absolute value bars like parentheses: evaluate what is inside first!

Quickfire Check

Discuss with your partner before voting on each value.

Partner Talk

1. Evaluate $|-17|$

?

2. Evaluate $-|24|$

?

3. True or False: $|-6| = |6|$

?

4. Distance between -3 and 0

?

Ready for the student worksheet! Keep your number lines handy

Zero Distance Student Worksheet

Distance Masters • Lesson 1

Zero Distance: Finding Absolute Value

Name: ___________________________

Date: ____________ Period: _____

Key Idea: The absolute value of a number is its distance from zero on a number line. Distance is always non-negative. Notation: $|x|$.

$|-a| = |a| = a$

1

Visual Number Lines

Draw an arc from 0 to each integer, state the number of steps, and complete the absolute value statement.

A. Plot and find the distance for -6:

Steps from 0: Statement: $|-6| =$

-8 -6 -4 -2 0 2 4 6 8

B. Plot and find the distance for +4:

Steps from 0: Statement: $|4| =$

-6 -4 -2 0 2 4 6

2

Evaluate Each Absolute Value

1. $|19|$ =

2. $|-31|$ =

3. $|0|$ =

4. $-|15|$ =

5. $|-8.5|$ =

6. $-|-42|$ =

7. $|-\frac{3}{4}|$ =

8. $-|0|$ =

3

Compare Values: Write $<$, $>$, or $=$

$|-12|$ $|12|$

$-15$ $|-9|$

$-|7|$ $-5$

$|-20|$ $|18|$

$-|-8|$ $|-8|$

$|0|$ $-|2|$

Distance Masters Series • Lesson 1 Student Worksheet Page 1 of 2

Lesson 1 • Applied Practice Name: ___________________________

4

Real-World Magnitude Problems

Model each scenario using an integer and its absolute value. Write complete explanations.

Badwater Basin in Death Valley sits at an elevation of -86 meters relative to sea level. Nearby Mount Whitney sits at +4,421 meters.

Depth/distance of Badwater Basin from sea level:

$|\text{Elevation}| =$

Which location has a greater distance from sea level?

Jordan has a bank balance of -$72, while Samantha has a balance of -$45.

Write absolute value expressions for the debt of each person:

Jordan: Samantha:

Who owes more money, and who has the greater account balance? Explain clearly:

5

Error Analysis: Help Detective Dani

Student Claim: "Since absolute value always turns negative signs into positive signs, $-|-16| = +16$!"

What is the error in the student's thinking?

What is the correct value of $-|-16|$?

Zero Distance Teacher Guide

Teacher Resource • Distance Masters

Lesson 1: Distance from Zero Guide

Middle School Math (Grades 6–7)

Unit 1 • 45–50 Minutes

Learning Objectives

Define absolute value as non-negative distance from zero on a number line.
Evaluate numerical absolute values including positive, negative, and zero.
Distinguish between $|-x|$ and $-|x|$ and interpret real-world contexts.

Suggested Pacing

5 min: Hook & Number Line Walk

15 min: Slides & Notation

20 min: Guided & Paired Work

10 min: Reflection & Exit Check

Critical Misconception Alert

Students frequently overgeneralize that "absolute value makes every sign positive". This causes errors on problems with leading negative signs outside the bars like $-|15|$ or $-|-16|$. Stress that absolute value acts as a grouping symbol first: $|-16| = 16$, then the outer negative yields $-16$.

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Student Worksheet Answer Key

Part 1: Number Lines

A. -6: 6 steps from 0; $|-6| = 6$.
B. +4: 4 steps from 0; $|4| = 4$.

Part 2: Core Values

1. 19 2. 31 3. 0 4. -15 5. 8.5 6. -42 7. 3/4 8. 0

Part 3: Comparisons

$|-12| \mathbf{=} |12|$ (both 12)

$-15 \mathbf{<} |-9|$ (-15 < 9)

$-|7| \mathbf{<} -5$ (-7 < -5)

$|-20| \mathbf{>} |18|$ (20 > 18)

$-|-8| \mathbf{<} |-8|$ (-8 < 8)

$|0| \mathbf{>} -|2|$ (0 > -2)

Part 4: Real-World Scenarios:

A. Death Valley: $|-86| = 86\text{ m}$. Mount Whitney has a greater distance from sea level ($|4,421| = 4,421\text{ m} > 86\text{ m}$).

B. Debt: Jordan's debt is $|-72| = \$72$; Samantha's debt is $|-45| = \$45$. Jordan owes more money ($72 > 45$), but Samantha has a greater balance ($-45 > -72$).

Part 5 & 6: Error Analysis & Reflection:

Part 5 Error: The student removed both negative signs at once. Inside $|-16| = 16$; the outside negative remains, resulting in $-16$.

Part 6 Reflection: Both $k$ and $-k$ are situated the same number of units away from zero on the number line in opposite directions. Since distance ignores direction, $|k| = |-k|$.

Magnitude Matchup Slides

Lesson 2 • Middle School Math

Distance Masters Series

Expressions & Magnitude

Learn how absolute value bars work inside the Order of Operations and calculate distance between any two numbers!

Grouping Symbols

Inside first, then bars

Distance Formula

$|a - b|$ on line

Change Scenarios

Temperature swings

Bars Are Grouping Symbols

In PEMDAS, absolute value bars act just like parentheses.

Rule #1

1 Simplify Inside

Complete all operations inside the bars first.

$|4 - 11| \rightarrow |-7|$

2 Take Distance

Apply the absolute value to make the inside positive.

$|-7| \rightarrow 7$

3 Finish Outer Math

Multiply, add, or subtract any numbers outside.

$5 + 7 = 12$

Never distribute into absolute value bars! Simplify the interior first. Inside → Absolute → Outside

Spot the Difference!

Where the bars sit completely changes the answer.

Look Closely

Case A: Subtraction Inside

$|7 - 12|$

• Compute inside first: $7 - 12 = -5$

• Apply absolute value: $|-5| = \mathbf{5}$

Result is 5 (positive distance)

Case B: Subtraction Outside

$|7| - |12|$

• Evaluate each bar first: $7$ and $12$

• Subtract outside: $7 - 12 = \mathbf{-5}$

Result is -5 (subtraction gives negative)

Conclusion: $|a - b| \neq |a| - |b|$. Always respect the grouping boundary!

Distance Between Two Numbers

Subtract the two values and take the absolute value.

Core Formula

The distance $d$ between point $a$ and point $b$ is:

$\text{Distance} = |a - b| = |b - a|$

Option 1: $|-4 - 6|$

$= |-10| = \mathbf{10\text{ units}}$

Start from -4 and subtract 6

Option 2: $|6 - (-4)|$

$= |6 + 4| = |10| = \mathbf{10\text{ units}}$

Start from 6 and subtract -4

Order doesn't matter! The absolute value guarantees distance is identical in both directions.

Real-Life: Temperature Swings

Crossing zero on a thermometer.

Context

The Problem:

At 6:00 AM, the temperature in Minneapolis was . By 2:00 PM, it rose to .

Magnitude Matchup Student Worksheet

Distance Masters • Lesson 2

Expressions and Magnitude

Name: ___________________________

Date: ____________ Period: _____

Order of Operations Rule: Treat bars $| \dots |$ as grouping symbols. Simplify the inner expression first, take the absolute value, then apply outside operations.

Inside → Bars → Outside

1

Evaluate Each Numerical Expression

Show your step-by-step simplification on each line provided.

A. $|4 - 11| + 6$

Step 1: Simplify inside =

Step 2: Take absolute value =

Final Answer:

B. $5 \cdot |-3| - 8$

Step 1: Absolute value of -3 =

Step 2: Multiply by 5 =

Final Answer:

C. $-|9 - 2| + 14$

Step 1: Simplify inside =

Step 2: Apply outside sign =

Final Answer:

D. $|16 - 25| \div 3$

Step 1: Inside subtraction =

Step 2: Divide by 3 =

Final Answer:

2

True or False? Justify with Math

Statement: $| -8 + 3 | = |-8| + |3|$

Left side evaluates to: _____ • Right side evaluates to: _____

True False

Statement: $| -5 \cdot 4 | = |-5| \cdot |4|$

Left side evaluates to: _____ • Right side evaluates to: _____

True False

3

Evaluate with Variables

Let $x = -5$ and $y = 3$. Substitute and calculate:

1. $|x| + |y|$

2. $|x - y|$

3. $-|2x + y|$

Distance Masters Series • Lesson 2 Student Worksheet Page 1 of 2

Lesson 2 • Geometric Distance & Modeling Name: ___________________________

4

Distance Between Two Coordinates: $d = |a - b|$

Calculate the distance between points $a$ and $b$. Verify by showing your work.

| Point $a$ | Point $b$ | Absolute Value Setup $|a - b|$ | Distance (Units) | | --- | --- | --- | --- | | -9 | 4 | $|-9 - 4| = |-13|$ | 13 | | -15 | -7 | $|-15 - (-7)| =$ | | | 18 | -6 | $|18 - (-6)| =$ |

| | -24 | 11 |

|

5

Word Problems • Practical Distance

A. Fargo Winter Temperature Swing

At midnight, the thermometer read -14°F. By noon the following day, the temperature was 23°F.

Write an absolute value expression for the temperature swing:

Magnitude Matchup Teacher Guide

Teacher Resource • Distance Masters

Lesson 2: Expressions & Magnitude Guide

Middle School Math (Grades 6–7)

Unit 2 • 45–50 Minutes

Learning Targets

Apply Order of Operations by simplifying within absolute value bars first.
Evaluate multi-step algebraic expressions with integer substitution.
Model and calculate the distance between any two coordinates using $|a - b|$.

Instructional Flow

7 min: Grouping Warm-up

13 min: Slide Examples

20 min: Paired Worksheet

10 min: Debrief & Mystery Teaser

Instructional Tip: The Bridge to Equations

Notice Part 6 on the worksheet introduces $|n - 4| = 9$. Use this as a natural conversational bridge to tomorrow's Lesson 3! Ask students: "If distance from 4 is 9 units, which way could you jump?" They will intuitively see you can jump 9 units right ($4 + 9 = 13$) or 9 units left ($4 - 9 = -5$). That sets up dual-branch solving without algebraic fear.

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Student Worksheet Answer Key

Part 1: Order of Operations

A. $|4 - 11| + 6 = |-7| + 6 = 7 + 6 = \mathbf{13}$
B. $5 \cdot |-3| - 8 = 5(3) - 8 = 15 - 8 = \mathbf{7}$
C. $-|9 - 2| + 14 = -|7| + 14 = -7 + 14 = \mathbf{7}$
D. $|16 - 25| \div 3 = |-9| \div 3 = 9 \div 3 = \mathbf{3}$

Part 2: True / False

Statement 1: False. Left side $|-5| = 5$, but right side $8 + 3 = 11$. ($5 \neq 11$)
Statement 2: True. Left side $|-20| = 20$, right side $5 \cdot 4 = 20$.

Part 3: Variable Values ($x=-5, y=3$)

1. $|-5| + |3| = 5 + 3 = \mathbf{8}$
2. $|-5 - 3| = |-8| = \mathbf{8}$
3. $-|2(-5) + 3| = -|-10 + 3| = -|-7| = \mathbf{-7}$

Part 4: Distance Table

• $(-15, -7)$: $|-15 - (-7)| = |-8| = \mathbf{8}$
• $(18, -6)$: $|18 - (-6)| = |24| = \mathbf{24}$
• $(-24, 11)$: $|-24 - 11| = |-35| = \mathbf{35}$

Part 5: Applications: A (Temperature): Expression $|23 - (-14)| = |37| = 37^\circ\text{F}$ swing. B (Football): Total ball travel $|-8| + |15| = 8 + 15 = 23\text{ yards}$. Net change $-8 + 15 = +7\text{ yards}$ (now on 42-yard line).

Equation Split Slides

Lesson 3 • Middle School Math

Distance Masters Series

Branching Equations

Why do absolute value equations produce two separate answers? Learn the branch method to find both!

Two Branches

Positive & negative

No Solution Trap

$|x| = -k$ is impossible

Target Tolerance

Real engineering margins

Why Do We Get Two Answers?

Distance looks in both directions from the center.

Intuition

Solve this mystery equation:

$|x| = 7$

"What numbers are exactly 7 units away from zero?"

Branch 1: Walk Right

$x = +7$

Check: $|7| = 7$ ✓

Branch 2: Walk Left

$x = -7$

Check: $|-7| = 7$ ✓

Whenever the right side is a positive number $c$, an absolute value equation splits into two branches!

Solving $|x + 3| = 8$

Drop the bars and split into positive and negative cases.

Algorithm

$|x + 3| = 8$

Positive Branch Inside = +8

$x + 3 = 8$

Subtract 3 from both sides:

$x = 5$

Negative Branch Inside = -8

$x + 3 = -8$

Subtract 3 from both sides:

$x = -11$

Solution Set: $x = 5$ or $x = -11$. Always check both back in the original equation!

Special Cases & Traps

Not every equation has two solutions!

Watch Out

Equal to a Negative 0 Solutions

$|x - 4| = -6$

Can distance ever be negative? NO! Stop immediately.

Answer: No Solution ($\varnothing$)

Equal to Zero 1 Solution

$|2x - 10| = 0$

Zero has no sign ($+0 = -0$). There is only one branch to solve!

$2x - 10 = 0 \implies x = 5$

Check the right-hand constant before splitting: positive (2), zero (1), or negative (0)!

Engineering Tolerance

How factory machines and bakers use absolute value.

Application

The Scenario: Cereal Box Weight

A machine fills cereal boxes to an ideal target weight of 450 grams. The scale allows an error (tolerance) of at most 6 grams.

Equation for the boundary weights:

$|\text{Weight} - \text{Target}| = \text{Tolerance}$

Equation Split Student Worksheet

Distance Masters • Lesson 3

Branching Equations

Name: ___________________________

Date: ____________ Period: _____

The Golden Branch Rule: When $|X| = c$ and $c > 0$, split into two linear equations: $X = c$ OR $X = -c$. If $c < 0$, there is no solution!

1

Warm-up: Determine the Solution(s)

A. $|x| = 14$

Two numbers 14 units from 0:

$x =$

B. $|m| = 0$

Numbers 0 units from 0:

$m =$

C. $|y| = -8$

Can distance be negative?

$y =$

2

Split into Two Branches and Solve

Set up the positive branch and negative branch. Show each step clearly.

1. $|x + 5| = 11$

Positive Branch:

$x + 5 = 11$

$x =$ _____

Negative Branch:

$x + 5 = -11$

$x =$ _____

Check solutions: $|(\dots) + 5| = 11$ ✓

2. $|w - 7| = 4$

Positive Branch:

$w - 7 = 4$

$w =$ _____

Negative Branch:

$w - 7 = -4$

$w =$ _____

Check solutions: $|(\dots) - 7| = 4$ ✓

3. $|2y - 6| = 12$

Positive Branch:

$2y - 6 = 12$

$y =$ _____

Negative Branch:

$2y - 6 = -12$

$y =$ _____

Check solutions: $|2(\dots) - 6| = 12$ ✓

4. $|p + 2| - 5 = 3$ (Isolate bars first!)

Isolate: $|p + 2| = 8$

$p + 2 = 8$

$p =$ _____

Negative Branch:

$p + 2 = -8$

$p =$ _____

Check solutions: $|(\dots) + 2| - 5 = 3$ ✓

3

Graph Solution Sets on Number Line

Graph the solution set of $|x - 1| = 4$ (which gives $x = 5$ and $x = -3$):

-5 -3 -1 1 3 5 7

Plot solid points at both solution coordinates. Notice that 1 is the exact midpoint!

Distance Masters Series • Lesson 3 Student Worksheet Page 1 of 2

Lesson 3 • Analysis & Engineering Tolerance Name: ___________________________

4

Identify Number of Solutions (0, 1, or 2)

State the number of solutions without solving fully, and explain your reasoning.

Equation Split Teacher Guide

Teacher Resource • Distance Masters

Lesson 3: Branching Equations Guide

Middle School Math (Grades 6–8)

Unit 3 • 45–50 Minutes

Learning Targets

Isolate absolute value bars and split into positive/negative linear branches.
Recognize equations with no solution ($|X| = -c$) and one solution ($|X| = 0$).
Model physical tolerances using $|\text{actual} - \text{target}| = \text{tolerance}$.

Instructional Flow

5 min: Two-Way Intuition Warm-up

15 min: Slide Walkthrough & Branching

20 min: Guided & Independent Worksheet

10 min: Unit Synthesis & Exit Ticket

Critical Pitfall: Premature Splitting

Students frequently attempt to branch before isolating the absolute value bars. For example, with $|p + 2| - 5 = 3$, a student might incorrectly write $p + 2 - 5 = 3$ and $p + 2 - 5 = -3$. Emphasize: The bars must stand alone on one side of the equal sign before opening the two branches!

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Student Worksheet Answer Key

Part 1: Warm-up Solutions

A. $|x| = 14 \implies \mathbf{x = 14 \text{ or } -14}$
B. $|m| = 0 \implies \mathbf{m = 0}$ (single solution)
C. $|y| = -8 \implies \mathbf{\text{No solution}}$ (distance ≥ 0)

Part 2: Two-Branch Solving

1. $x + 5 = 11 \implies x = 6$; $x + 5 = -11 \implies \mathbf{x = -16}$
2. $w - 7 = 4 \implies w = 11$; $w - 7 = -4 \implies \mathbf{w = 3}$
3. $2y - 6 = 12 \implies 2y = 18 \implies y = 9$; $2y - 6 = -12 \implies 2y = -6 \implies \mathbf{y = -3}$
4. $|p + 2| = 8 \implies p = 6\text{ or }\mathbf{p = -10}$

Part 3: Solution Graphing

Solid plotted points at $\mathbf{-3}$ and $\mathbf{5}$. Note center is $1$, and both dots are exactly 4 units away from 1.

Part 4: Solution Count

• $|3x - 12| = 0 \implies \mathbf{1\text{ sol}}$ ($3x = 12 \implies x = 4$)
• $|y - 15| = 22 \implies \mathbf{2\text{ sol}}$ (positive constant)
• $|m + 6| + 7 = 3 \implies |m + 6| = -4 \implies \mathbf{0\text{ sol}}$

Part 5: Tolerance Modeling: A (Flight): Equation $|h - 33,000| = 400$. Min altitude $= 33,000 - 400 = \mathbf{32,600\text{ ft}}$; Max $= 33,000 + 400 = \mathbf{33,400\text{ ft}}$. Equation $|T - 238| = 3$. Min temp $= 238 - 3 = \mathbf{235^\circ\text{F}}$; Max $= 238 + 3 = \mathbf{241^\circ\text{F}}$.

6

Reflection • Number Line Reasoning

Suppose $k$ is a mystery integer. Explain why $|k| = |-k|$ is always true regardless of whether $k$ is positive, negative, or zero. Use the word distance in your explanation:

Distance Masters Series • Lesson 1 Student Worksheet Page 2 of 2

Distance Masters Series • Lesson 1 Facilitation Guide Middle School Instructional Resource

-12°F

18°F

How many degrees did the temperature change?

The Setup:

$|18 - (-12)| = |18 + 12|$

$= |30| = \mathbf{30^\circ\text{F}}$

The temperature traveled 12 degrees to reach zero, then 18 degrees more into positive numbers!

Distance across zero is the sum of their distances from zero: $|-12| + |18| = 12 + 18 = 30^\circ\text{F}$.

Showdown Challenge

Solve each expression on your whiteboards in 30 seconds!

Practice

1. $|3 - 9| + 4$

?

2. $2 \cdot |-8| - 6$

?

3. $-|15 - 5|$

?

4. Distance from -7 to 8

?

Mastery unlocked! Ready for the worksheet practice. Order of Operations rules apply!

Calculate total degrees changed:

B. Football Field Position

A football team begins at the 35-yard line. On 1st down they lose 8 yards (represented as -8). On 2nd down they gain 15 yards (+15).

Total distance traveled by the ball on both plays $(|-8| + |15|)$:

Net change in yardage from original line $(-8 + 15)$:

6

Brain Teaser: The Mystery Number

"A secret integer $n$ is located on the number line. The distance between $n$ and 4 is exactly 9 units. What are the two possible values of $n$?"

Write equation: $|n - 4| = 9$

Two possible values for $n$:

Distance Masters Series • Lesson 2 Student Worksheet Page 2 of 2

Part 6: Mystery Number: Equation: $|n - 4| = 9$. Either $n - 4 = 9 \implies n = 13$, or $n - 4 = -9 \implies n = -5$. Both -5 and 13 are 9 units away from 4!

Distance Masters Series • Lesson 2 Facilitation Guide Middle School Instructional Resource

$|w - 450| = 6$

Minimum Allowable Weight $450 - 6 = 444\text{ grams}$

Maximum Allowable Weight $450 + 6 = 456\text{ grams}$

Formula structure: $|\text{Actual} - \text{Ideal}| = \text{Tolerance}$.

Unit Mastery Checklist

Three golden rules to remember when solving equations.

Summary

1. Isolate First

Make sure the absolute value bars are completely alone before splitting.

$|x| = c$

2. Dual Branches

Create one positive branch and one negative branch.

$+(\dots)$ and $-(\dots)$

3. Always Check

Plug both answers into the original equation to verify.

Substitute & Verify

You are now a Distance Master! Complete the final worksheet challenge

Equation	# of Solutions	Mathematical Justification
\(	x + 9	= -4\)
\(	3x - 12	= 0\)

| | $|y - 15| = 22$ |

|

| | $|m + 6| + 7 = 3$ |

|

5

Engineering & Science Tolerance Word Problems

A. Commercial Airline Flight Corridor

A commercial airplane is assigned a cruising altitude of 33,000 feet. Due to turbulence, air traffic control allows the altitude $h$ to vary by up to 400 feet.

Write an absolute value equation for the boundary altitudes: $|h - \text{______}| = \text{______}$

Minimum allowable altitude:

Maximum allowable altitude:

B. Candy Making: Soft Ball Stage

When making caramel, sugar syrup must stay within 3 degrees of 238°F.

Write absolute value equation for temperature $T$:

Find the minimum & maximum temperatures:

6

Unit Capstone Synthesis

Why must you isolate the absolute value bars before splitting an equation into two branches? (Hint: consider $|x| + 5 = 2$)

Distance Masters Series • Lesson 3 Student Worksheet Page 2 of 2

B (Candy):

Part 6: Capstone Synthesis: The definition $|X| = c$ splitting into $X = \pm c$ only works when the absolute value is isolated. In $|x| + 5 = 2$, isolating yields $|x| = -3$, showing there are actually zero solutions. Splitting before isolating produces false extraneous answers.

Distance Masters Series • Lesson 3 Facilitation Guide Middle School Instructional Resource

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