Phase 1: Briefing
Register as a detective and review your case vocabulary.
🕵️ Your Mission, Detective
Welcome to the Equations & Inequalities Division. As a newly recruited detective, you will use your math skills to solve one-step equations, write and graph inequalities, and crack multi-step cases. Each task is a piece of evidence that brings you closer to solving the mystery. Complete all case files to earn your detective badge and prove you have what it takes to bring justice to the city!
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Task 1: Detective Registration
Register with the department by providing your information and
naming your detective agency.
Choose a creative name for your investigation agency.
📚 Support: Think of a name that relates
to math concepts like equations, variables, or solving. For
example: “The Variable Detectives” or “Balance
Squad.”
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Task 2: Vocabulary Match
Every detective needs to know the language. Match each term with
its correct definition by entering the number (1–4) next to
each letter.
Definitions
(1) A letter that represents an unknown
value
(2) A math sentence with an = sign
(3) A math sentence with <, >, ≤, or ≥
(4) A value that makes a statement true
(2) A math sentence with an = sign
(3) A math sentence with <, >, ≤, or ≥
(4) A value that makes a statement true
📚 Support: An equation uses an
equals sign (=). An inequality uses symbols like <,
>, ≤, ≥. A variable is a letter (like x or n). A
solution is the answer that makes the statement true.
🌟 Challenge: Can you write a sentence
using all four vocabulary words? Write it below.
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Task 3: Case Classification
Classify each mathematical statement as an
Equation (EQ) or an
Inequality (IN). Type “EQ” or
“IN” for each.
| Case | Statement | Type (EQ or IN) |
|---|---|---|
| (a) | 3x = 21 | |
| (b) | n + 5 > 12 | |
| (c) | y / 4 = 8 | |
| (d) | 2m ≤ 16 | |
| (e) | p − 7 = 3 |
📚 Support: Look for the symbol in the
middle. If it is an = sign, it is an
Equation (EQ). If it is <, >, ≤, or
≥, it is an Inequality (IN).
🌟 Challenge: Write your own equation AND
inequality and classify each.
Phase 2: Evidence Research
Solve one-step equations to uncover evidence for your cases.
🔍 Gathering Evidence
Each equation below represents a clue in an active investigation. Solve each one-step equation to unlock the evidence. Remember: what you do to one side, you must do to the other side to keep the balance!
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Task 4: The Missing Deposit — Addition Equation
A bank reported that an account had a deposit of $28 added,
bringing the total to $75. How much was originally in the account?
x + 28
Left Side
=
75
Right Side
📚 Support: To solve
x + 28 = 75, subtract 28 from both
sides: x = 75 − 28 = 47.
Addition equations are solved with subtraction (inverse
operation).
Since 28 is being added to x, use the inverse operation
(subtraction). Subtract 28 from both sides.
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Task 5: The Stolen Evidence — Subtraction Equation
A detective had a collection of evidence items but 14 went
missing. After the theft, 23 items remained. How many items were
there originally?
n − 14
Left Side
=
23
Right Side
📚 Support: To solve
n − 14 = 23, add 14 to both
sides: n = 23 + 14 = 37. Subtraction
equations are solved with addition (inverse operation).
Since 14 is being subtracted from n, use the inverse operation
(addition). Add 14 to both sides.
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Task 6: The Counterfeit Ring — Multiplication Equation
Investigators discovered 5 identical counterfeit money bags worth
a total of $145. How much was in each bag?
5m
Left Side
=
145
Right Side
📚 Support: To solve
5m = 145, divide both sides by 5:
m = 145 ÷ 5 = 29. Multiplication
equations are solved with division.
5m means “5 times m.” To undo multiplication, divide
both sides by 5.
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Task 7: The Split Reward — Division Equation
A reward was split equally among 6 detectives. Each detective
received $48. What was the total reward?
r / 6
Left Side
=
48
Right Side
📚 Support: To solve
r / 6 = 48, multiply both sides by
6: r = 48 × 6 = 288. Division
equations are solved with multiplication.
🌟 Challenge: Write a real-world scenario
for the equation p / 3 = 17.
r / 6 means “r divided by 6.” To undo division,
multiply both sides by 6.
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Task 8: The Partial Clue — Fractional Equation
A coded message revealed that one-third of the stolen items equals
24. How many items were stolen in total?
x / 3
Left Side
=
24
Right Side
📚 Support: “One-third of x”
means x / 3. Multiply both sides by
3: x = 24 × 3 = 72.
x / 3 = 24 is the same as ⅓ of x = 24. Multiply both
sides by 3 to find the whole amount.
Phase 3: Case Planning
Write, solve, and graph inequalities to plan your investigation.
📈 Planning the Investigation
Inequalities help detectives set boundaries and constraints. In this phase, you will write inequalities from real-world scenarios, learn to graph them on number lines, and understand open vs. closed circles.
Reference: Graphing x > 5 on a Number Line
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Task 9: Write the Inequality — Height Requirement
To join the detective academy, a candidate must be
at least 54 inches tall. Write an inequality
using h for height.
📚 Support: “At least” means
greater than or equal to, which uses the ≥
symbol. Written as: h >= 54.
“At least 54” means 54 or more. The candidate can be
54 inches tall (exactly) or taller. Which symbol includes the
value itself?
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Task 10: Graph the Inequality — Suspect Age Range
The suspect is older than 25. This means
x > 25. Describe how you would
graph this on a number line. Is the circle open or closed? Which
direction does the arrow point?
📚 Support:
Open circle = the value is NOT included (> or
<). Closed circle = the value IS included
(≥ or ≤). Since x > 25 does NOT include 25 itself, we use
an open circle. The arrow points
right (toward bigger numbers).
> means “greater than” but NOT equal to. So 25
itself is not a solution. What kind of circle represents
“not included”?
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Task 11: Solve and Graph — The Budget Constraint
The department has a $200 budget for an investigation. Equipment
costs $40 upfront, and each hour of surveillance costs $20. How
many hours of surveillance can the team afford?
40 + 20h ≤ 200
40 + 20h ≤ 200
📚 Support:
Step 1: Subtract 40 from both sides:
20h ≤ 160. Step 2: Divide both
sides by 20: h ≤ 8. The team can
afford up to 8 hours of surveillance.
🌟 Challenge: What if the budget was $300
instead? How many hours could the team afford then? Show your
work.
This is a two-step inequality. First undo the addition (+40),
then undo the multiplication (×20). Keep the ≤ sign
pointing the same way.
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Task 12: Open vs. Closed Circle
For each inequality, decide whether the boundary point uses an
open circle or a closed circle.
| Case | Inequality | Circle Type |
|---|---|---|
| (a) | x > 7 | |
| (b) | y ≤ 3 | |
| (c) | n ≥ 10 | |
| (d) | p < 5 |
📚 Support:
Open circle is used for < and > (the value
is NOT part of the solution). Closed circle is
used for ≤ and ≥ (the value IS part of the solution).
Remember: if the symbol has a line underneath (≤ or ≥), it
means “or equal to,” so the endpoint is
included = closed circle.
Phase 4: Investigation Build
Apply your skills to complex, real-world detective cases.
🔨 Building the Case
Now it is time to apply everything you have learned. These cases involve real-world scenarios that require equations and inequalities to solve. Think carefully, show your work, and build an airtight case!
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Task 13: The Jewelry Heist
Investigators found that 4 identical jewelry cases were stolen.
The total value of the stolen goods is $2,340. How much is each
case worth?
4c = 2340
4c = 2340
4c
Left Side
=
2340
Right Side
📚 Support: To solve
4c = 2340, divide both sides by 4:
c = 2340 ÷ 4 = 585. Each case is
worth $585.
4c means 4 times c. To find one case, divide the total by 4.
Try: 2340 ÷ 4.
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Task 14: The Speeding Suspect
A suspect drove 324 miles in 6 hours. What was the suspect’s
average speed?
6s = 324
Then: The speed limit is 60 mph. Was the suspect speeding?
6s = 324
Then: The speed limit is 60 mph. Was the suspect speeding?
6s
Left Side
=
324
Right Side
📚 Support: The formula is
distance = speed × time, or
d = s × t. So
6s = 324 ⇒
s = 324 ÷ 6 = 54 mph. Since 54
< 60, the suspect was NOT speeding.
First find the speed by dividing 324 by 6. Then compare your
answer to 60. Is it more or less than 60?
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Task 15: The Ransom Note — Two-Step Equation
A ransom note demands that half of the stolen amount plus $120
equals $400. What was the total stolen amount?
x/2 + 120 = 400
x/2 + 120 = 400
x/2 + 120
Left Side
=
400
Right Side
📚 Support: This is a
two-step equation.
Step 1: Subtract 120 from both sides: x/2 = 280.
Step 2: Multiply both sides by 2: x = 560.
Step 1: Subtract 120 from both sides: x/2 = 280.
Step 2: Multiply both sides by 2: x = 560.
🌟 Challenge: Write your own two-step
equation with a real-world scenario. Include the equation and the
solution.
Two-step equations: undo addition/subtraction first, then undo
multiplication/division. Think of “reverse PEMDAS.”
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Task 16: The Witness Timeline
A witness saw the suspect at the scene for
more than 3 hours but no more than 7 hours. Write
this as a compound inequality using t for time,
then name three integers that satisfy it.
3 < t ≤ 7
3 < t ≤ 7
📚 Support: “More than 3”
means t > 3 (open circle at 3, not included). “No more
than 7” means t ≤ 7 (closed circle at 7, included).
Together: 3 < t ≤ 7. Valid
integers: 4, 5, 6, 7.
“More than” = strictly greater than (>).
“No more than” = less than or equal to (≤). Which
whole numbers are between 3 and 7 following these rules?
Phase 5: Portfolio
Summarize your findings, reflect, and submit your case report.
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Task 17: Case Summary Report
Write a summary of your detective work. Use at least
3 vocabulary words from the unit (equation,
inequality, variable, solution, inverse operation, open circle,
closed circle). Explain what you learned and how you solved the
cases.
(minimum 20 characters)
📚 Support: Try to include these terms:
equation, inequality,
variable, solution,
inverse operation. Example start: “In this
project, I solved equations by using inverse operations to find
the value of the variable...”
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Task 18: Create Your Own Case
Design your own detective case! Your case must include:
• An equation that needs to be solved
• An inequality that needs to be written and graphed
• A description of what the number line graph looks like (minimum 20 characters)
• An equation that needs to be solved
• An inequality that needs to be written and graphed
• A description of what the number line graph looks like (minimum 20 characters)
📚 Support: Think of a real-world
situation (money, distance, time, age). Write the equation, solve
it, then create a related inequality. Describe if the number line
has an open or closed circle and which direction the arrow points.
🌟 Challenge: Include a two-step equation
in your case and a compound inequality (like Task 16). Show the
full solution for both.
★
Self-Assessment Rubric
Rate yourself in each category. Click a score for each row.
1. Solving One-Step Equations
1
I need a lot of help solving one-step equations.
2
I can solve some one-step equations with support.
3
I can solve one-step equations independently.
4
I can solve equations and explain my reasoning clearly.
2. Writing & Graphing Inequalities
1
I need help writing and graphing inequalities.
2
I can write or graph inequalities with support.
3
I can write and graph inequalities on my own.
4
I can write, graph, and explain inequalities with confidence.
3. Real-World Application
1
I struggle to connect math to real-world problems.
2
I can sometimes see math in real situations.
3
I can apply equations and inequalities to real-world cases.
4
I can create my own real-world problems and solve them.
4. Mathematical Communication
1
I have trouble explaining my math thinking.
2
I can partially explain my steps.
3
I clearly show my work and explain my steps.
4
I communicate with math vocabulary and clear reasoning.
📝
Reflection Journal
Answer each reflection question thoughtfully.
Your Detective Score
0
out of 100
Equation Solver
Inequality Expert
Number Line Pro
Vocab Master
Case Writer