5.6 Unit Review & Assessment
Comprehensive review of all Unit 5 topics — graphing, writing, special cases, and real-world applications — plus a full unit assessment.
Why This Matters
Systems of inequalities connect algebra to real-world optimization — a concept that appears in economics, engineering, and data science. Reviewing this unit prepares you for linear programming and advanced graphing in Precalculus.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Unit 5 Concept Map
Systems of Inequalities
Two or more inequalities solved simultaneously
Graphing
Boundary lines + shading
Feasible Region
Overlap of all shaded areas
Writing from Context
Words → Variables → Inequalities
Special Cases
No solution / Unbounded
Applications
Real-world constraints & modeling
Chapter-by-Chapter Summary
Chapter 01
Introduction to Systems of Inequalities
- ✓A system of inequalities is two or more inequalities with the same variables
- ✓A solution is any ordered pair (x, y) that satisfies ALL inequalities simultaneously
- ✓Graph each inequality separately; the feasible region is the overlap
- ✓Test points to verify which side of the boundary line to shade
Chapter 02
Graphing Systems of Inequalities
- ✓Rewrite each inequality in slope-intercept form (y = mx + b)
- ✓Solid boundary line for ≤ or ≥; dashed boundary line for < or >
- ✓Shade above the line for y > or y ≥; shade below for y < or y ≤
- ✓The feasible region is where all shaded areas overlap
- ✓Vertices are found by solving pairs of boundary equations simultaneously
Chapter 03
Writing Systems of Inequalities
- ✓Identify key quantities and assign variables (x and y)
- ✓Translate constraint language: "at most" → ≤, "at least" → ≥, "less than" → <, "more than" → >
- ✓Include non-negativity constraints (x ≥ 0, y ≥ 0) for real-world problems
- ✓Read the graph: determine slope and y-intercept of each boundary line
- ✓Determine inequality direction from the shaded region
Chapter 04
Special Cases: No Solution & Unbounded Regions
- ✓No solution: parallel boundary lines with shading in opposite directions — no overlap
- ✓Unbounded: feasible region extends infinitely in at least one direction
- ✓Bounded: feasible region is an enclosed polygon with finite vertices
- ✓Minimum 3 non-parallel constraints needed to create a bounded region
- ✓Real-world systems often include x ≥ 0 and y ≥ 0 to bound the region
Chapter 05
Applications of Systems of Inequalities
- ✓6-step modeling process: Read → Define → Write → Graph → Identify → Interpret
- ✓Budget constraints: cost₁·x + cost₂·y ≤ budget
- ✓Labor/time constraints: time₁·x + time₂·y ≤ total time
- ✓Test feasibility: substitute a point into all inequalities
- ✓Interpret the feasible region in context — every point is a valid solution
Key Vocabulary Review
Formula & Strategy Reference Sheet
| Inequality Symbol | Meaning | Boundary Line | Shade Direction |
|---|---|---|---|
| y > mx + b | y is greater than | Dashed | Above the line |
| y ≥ mx + b | y is greater than or equal to | Solid | Above the line |
| y < mx + b | y is less than | Dashed | Below the line |
| y ≤ mx + b | y is less than or equal to | Solid | Below the line |
| Keyword / Phrase | Inequality Symbol |
|---|---|
| at most, no more than, maximum, up to | ≤ |
| at least, no less than, minimum, or more | ≥ |
| less than, fewer than, under, below | < |
| more than, greater than, above, exceeds | > |
| exactly, equal to, is | = |
Decision Tree — Classifying the Feasible Region
Shading in opposite directions?
Parallel boundary lines
NO SOLUTION
Empty feasible region
Shading in same direction?
Parallel boundary lines
UNBOUNDED
Infinite feasible region
3+ non-parallel lines?
Intersecting boundaries
BOUNDED
Enclosed polygon region
Solid vs. Dashed Boundary Lines
Solid line: y ≤ x + 1 (≤ includes boundary)
Boundary is part of the solution
Dashed line: y > x + 1 (> excludes boundary)
Boundary is NOT part of the solution
Feasible Region Types — Visual Review
Bounded Region
Enclosed polygon
Unbounded Region
Extends infinitely upward
No Solution
No overlap — empty region
Feasible Region Checklist
I rewrote each inequality in slope-intercept form (y = mx + b)
I drew solid lines for ≤ and ≥ inequalities
I drew dashed lines for < and > inequalities
I shaded above the line for y > and y ≥
I shaded below the line for y < and y ≤
I identified the feasible region as the overlap of all shaded areas
I tested a point in the feasible region to verify it satisfies all inequalities
I classified the region as bounded, unbounded, or no solution
I found all vertices by solving pairs of boundary equations
I interpreted the feasible region in context (for word problems)
Common Mistakes Review
Student Error
Student writes y ≤ x + 2 for "y is greater than x + 2"
Correct Answer
y ≥ x + 2
Explanation
"Greater than or equal to" means y is on the larger side — use ≥, not ≤. The inequality symbol points toward the smaller value.
Student Error
Student draws a solid line for y > 3x − 1
Correct Answer
Draw a dashed line
Explanation
Strict inequalities (< and >) use dashed boundary lines because the boundary itself is NOT included in the solution set.
Student Error
Student shades below the line for y ≥ 2x + 1
Correct Answer
Shade above the line
Explanation
For y ≥ (greater than or equal to), shade the region ABOVE the boundary line — those are the points where y is at least as large as 2x + 1.
Student Error
Student says "parallel lines always mean no solution"
Correct Answer
Parallel lines with shading in the SAME direction → unbounded; OPPOSITE directions → no solution
Explanation
The direction of shading determines the outcome. Two parallel lines can produce either no solution or an unbounded region depending on which sides are shaded.
Student Error
Student writes x + y ≤ 10 for "at least 10 items total"
Correct Answer
x + y ≥ 10
Explanation
"At least" means the minimum is 10 — the total must be 10 or more. Use ≥ for "at least" and ≤ for "at most."
Graph Interpretation Guide
Identify the feasible region
Green point is in the feasible region
Vertex identification
Vertex: solve y=−x+4 and y=x → (2,2)
Common Mistakes
Using a solid boundary line for a strict inequality (< or >) when graphing.
Dashed line for < or >. Solid line for ≤ or ≥.
Shading each inequality separately without finding the overlap — the feasible region is the intersection.
Graph both boundaries, shade each region, then identify and darken only the overlapping area.
Writing = instead of ≤ or ≥ for real-world constraints like budgets or minimums.
'At most' → ≤. 'At least' → ≥. Use = only when the value must be exactly that amount.
Forgetting non-negativity constraints (x ≥ 0, y ≥ 0) for real-world quantities.
Physical quantities can't be negative. Always include x ≥ 0 and y ≥ 0 unless the context explicitly allows negatives.
Mixed Review Problems
Guided Practice Video: Unit 5 Review
Watch the Unit 5 review walkthrough, then work through the mixed review problems below.
Video by Sang Real Math
Watch on YouTube ↗Part 1 — Graphing Systems (Problems 1–6)
Graph the system: y ≤ 2x + 1 and y > −x + 3. Shade the feasible region.
Hint: Draw y = 2x + 1 as a solid line (shade below) and y = −x + 3 as a dashed line (shade above). The feasible region is the overlap.
Graph: y ≥ x − 2 and y ≤ −x + 4. Identify all vertices of the feasible region.
Hint: Find the intersection of y = x − 2 and y = −x + 4 by setting them equal: x − 2 = −x + 4 → x = 3, y = 1. Vertex: (3, 1).
Graph: x + y ≤ 6 and x ≥ 1 and y ≥ 0. Classify the feasible region and find all vertices.
Hint: Three constraints form a bounded triangular region. Vertices are at intersections of pairs of boundary lines.
Graph: y > 2x − 1 and y > 2x + 3. Classify the system.
Hint: Both lines have slope 2 (parallel). Check shading direction: both shade above → unbounded; opposite → no solution.
Graph: y ≤ −x + 5 and y ≤ x + 1 and y ≥ 0. Find all vertices.
Hint: Find intersections: y = −x + 5 and y = x + 1 → (2, 3); y = −x + 5 and y = 0 → (5, 0); y = x + 1 and y = 0 → (−1, 0).
Graph: 2x + y ≤ 8 and x − y ≥ −2 and x ≥ 0. Shade the feasible region.
Hint: Rewrite in slope-intercept form: y ≤ −2x + 8 and y ≤ x + 2. Graph both with solid lines and shade the overlap.
Part 2 — Writing Systems (Problems 7–12)
Write the system of inequalities: "x is at most 5 and y is at least 2 and the sum of x and y is less than 10."
Hint: Translate each phrase: "at most 5" → x ≤ 5; "at least 2" → y ≥ 2; "sum less than 10" → x + y < 10.
Write the system from the graph: solid line y = x + 2 (shaded below) and dashed line y = −2x + 4 (shaded above).
Hint: Solid line shaded below → y ≤ x + 2. Dashed line shaded above → y > −2x + 4.
A student earns $10/hr tutoring (x) and $8/hr babysitting (y). She wants to earn at least $80 and work at most 10 hours. Write the system.
Hint: Earnings: 10x + 8y ≥ 80. Hours: x + y ≤ 10. Non-negativity: x ≥ 0, y ≥ 0.
A farmer plants corn (x acres) and wheat (y acres). He has 20 acres total and must plant at least 5 acres of each. Write the system.
Hint: Total: x + y ≤ 20. Minimums: x ≥ 5 and y ≥ 5.
Write the system from the graph: two parallel solid lines y = 3x + 1 and y = 3x − 4, with the region between them shaded.
Hint: The region between two parallel lines: y ≤ 3x + 1 (below upper line) and y ≥ 3x − 4 (above lower line).
A bakery makes muffins (x) and cookies (y). Each muffin needs 2 cups of flour; each cookie needs 1 cup. They have 12 cups of flour. They must make at least 3 muffins and at least 4 cookies. Write the complete system.
Hint: Flour: 2x + y ≤ 12. Minimums: x ≥ 3 and y ≥ 4.
Part 3 — Special Cases (Problems 13–16)
Classify: y ≥ 2x + 5 and y ≤ 2x − 1. Explain your reasoning.
Hint: Both lines have slope 2 (parallel). y ≥ 2x + 5 shades above the upper line; y ≤ 2x − 1 shades below the lower line. No overlap → no solution.
Classify: y ≤ x + 4 and y ≤ −x + 2. Is the region bounded or unbounded?
Hint: Two non-parallel lines both shaded below. The overlap extends downward infinitely → unbounded.
Add one constraint to y ≥ x − 1 and y ≥ −x + 1 to make the feasible region bounded.
Hint: The current system is unbounded (both shade above, region extends upward). Add y ≤ 4 or any horizontal upper bound to close the region.
A system has constraints x ≥ 0, y ≥ 0, and x + y ≤ 8. How many vertices does the feasible region have? Find them all.
Hint: Three constraints form a triangle. Vertices: (0, 0), (8, 0), and (0, 8).
Part 4 — Real-World Applications (Problems 17–20)
A gym offers yoga classes ($12 each) and spin classes ($15 each). A member has a budget of at most $90 and wants to attend at least 6 classes total. Write the system, graph it, and test whether (3, 4) is feasible.
Hint: Budget: 12x + 15y ≤ 90. Classes: x + y ≥ 6. Test (3, 4): 12(3) + 15(4) = 36 + 60 = 96 > 90 → NOT feasible.
A factory makes product A (3 hrs each) and product B (2 hrs each). It has 18 labor hours and must make at least 2 of each product. Write the system and identify two feasible production combinations.
Hint: Labor: 3x + 2y ≤ 18. Minimums: x ≥ 2 and y ≥ 2. Try (2, 6): 6 + 12 = 18 ✓. Try (4, 3): 12 + 6 = 18 ✓.
A nutritionist designs a meal with food X (100 cal, 5g protein per serving) and food Y (80 cal, 10g protein per serving). The meal must have at most 400 calories and at least 30g protein. Write and graph the system.
Hint: Calories: 100x + 80y ≤ 400 → 5x + 4y ≤ 20. Protein: 5x + 10y ≥ 30 → x + 2y ≥ 6.
A school store sells pencils ($0.50) and notebooks ($2.00). A student has at most $8 and wants at least 4 items total. Write the system, graph it, and name two feasible combinations.
Hint: Cost: 0.5x + 2y ≤ 8. Items: x + y ≥ 4. Try (8, 0): 4 + 0 = 4 ✓ and 0.5(8) = 4 ≤ 8 ✓. Try (4, 2): 2 + 4 = 6 ≤ 8 ✓.
Multiple Choice Review (10 Questions)
1. Which point is in the feasible region of y ≤ x + 3 and y ≥ −x + 1? (A) (0, 5) (B) (2, 1) (C) (−1, 3) (D) (3, 7)
2. A system has two parallel boundary lines with shading in opposite directions. The feasible region is: (A) Bounded (B) Unbounded (C) No solution (D) A single point
3. Which inequality is graphed with a DASHED boundary line? (A) y ≤ 3x + 1 (B) y ≥ −x + 2 (C) y > 2x − 4 (D) y ≤ x + 5
4. The system x + y ≤ 10, x ≥ 2, y ≥ 3 has how many vertices? (A) 2 (B) 3 (C) 4 (D) 5
5. "At most 15 items" is correctly written as: (A) x > 15 (B) x < 15 (C) x ≥ 15 (D) x ≤ 15
6. Which system has NO solution? (A) y ≥ x+1 and y ≤ x+5 (B) y ≤ 2x+3 and y ≥ 2x+7 (C) y ≥ x and y ≤ −x+4 (D) y < x+2 and y > −x+1
7. To create a BOUNDED feasible region, the minimum number of non-parallel constraints needed is: (A) 1 (B) 2 (C) 3 (D) 4
8. A student has $20 for apples ($1 each) and oranges ($2 each). The constraint is: (A) x + 2y ≥ 20 (B) x + 2y ≤ 20 (C) 2x + y ≤ 20 (D) x + y ≤ 20
9. The feasible region of y ≥ 0, x ≥ 0, and x + y ≤ 5 is: (A) Unbounded (B) No solution (C) Bounded triangle (D) Bounded rectangle
10. Which ordered pair satisfies BOTH y > 2x − 1 AND y ≤ −x + 5? (A) (3, 2) (B) (0, 4) (C) (4, 1) (D) (2, 5)
Short Response Review (10 Questions)
11. Write the system of inequalities for: "x is positive, y is positive, and their sum is at most 8."
12. A graph shows a solid line y = 2x − 3 with shading above it and a dashed line y = −x + 4 with shading below it. Write the system.
13. Test whether (1, 3) is in the feasible region of: 2x + y ≤ 8 and x + 3y ≥ 10. Show all work.
14. Classify the system y ≤ 3x + 2 and y ≥ 3x − 5. Explain your reasoning.
15. Find the vertex of the feasible region formed by y = x + 1 and y = −x + 7.
16. A company needs at least 10 units of product A and at least 5 units of product B, with a total of at most 20 units. Write the complete system including non-negativity constraints.
17. Explain the difference between a bounded and an unbounded feasible region. Give one real-world example of each.
18. A student graphs y ≥ x + 2 and shades below the line. Identify the error and explain the correction.
19. Write a real-world scenario that could be modeled by: x + y ≤ 12 and 3x + 5y ≤ 48 and x ≥ 0 and y ≥ 0.
20. The system x ≥ 0, y ≥ 0, x + y ≤ 6, and x − y ≤ 2 forms a bounded region. Find all four vertices.
Challenge Problems
C1. A system has three constraints forming a triangle with vertices at (0, 0), (4, 0), and (0, 6). Write the three inequalities that define this bounded region.
C2. Graph the system: y ≥ |x| − 2 (hint: this is two inequalities: y ≥ x − 2 for x ≥ 0 and y ≥ −x − 2 for x < 0). Describe the feasible region.
C3. A system has constraints 2x + 3y ≤ 12, x − y ≥ −1, and x + 2y ≥ 4. Find all vertices of the feasible region by solving each pair of boundary equations.
C4. A company makes two products. Product A requires 3 hours of machine time and 2 hours of labor. Product B requires 1 hour of machine time and 4 hours of labor. The company has 15 machine hours and 20 labor hours. Write the system, graph it, and find all vertices.
C5. Prove that the point (3, 2) is a vertex of the feasible region formed by y ≤ x + 1, y ≥ −x + 3, and y ≥ 0 by showing it satisfies all three inequalities and lies on two boundary lines.
Real-World Modeling Problems
Modeling Organizer
Context
A student has $30 to spend on books ($5 each) and supplies ($3 each). She wants at least 2 books and at least 4 supplies.
Variables
x = number of books; y = number of supplies
System of Inequalities
- 5x + 3y ≤ 30
- x ≥ 2
- y ≥ 4
- x ≥ 0, y ≥ 0
Feasible Region
Bounded region; all combinations of books and supplies within budget and minimums
Modeling Organizer
Context
A trainer schedules cardio sessions (45 min each) and strength sessions (60 min each). She has at most 300 minutes per week and must schedule at least 2 of each type.
Variables
x = cardio sessions; y = strength sessions
System of Inequalities
- 45x + 60y ≤ 300
- x ≥ 2
- y ≥ 2
- x ≥ 0, y ≥ 0
Feasible Region
Bounded region; all valid weekly training schedules
Modeling Organizer
Context
A restaurant uses chicken ($4/lb) and beef ($7/lb). The daily budget is at most $56. They need at least 3 lbs of chicken and at least 2 lbs of beef.
Variables
x = lbs of chicken; y = lbs of beef
System of Inequalities
- 4x + 7y ≤ 56
- x ≥ 3
- y ≥ 2
- x ≥ 0, y ≥ 0
Feasible Region
Bounded region; all valid daily ingredient combinations