Business Pivot Slides Algebra 2: Business Consulting
BUSINESS
PIVOT
Analyzing "No Solution" Systems in the Real World
Strategic Analysis Unit
Morning Huddle
"Have you ever had a plan that failed because you didn't have enough time or money?"
Example 1
"I wanted to go to the concert, but tickets were $100 and I only had $40."
Example 2
"I wanted to finish my project and play video games, but I only had 2 hours."
In math, we call these Constraints .
Case Study: Furniture Store
Timestamp: 3:49 - 6:45
Embedded media
Key Data
Sofa: 21 sq ft | $200 profit
Recliner: 9 sq ft | $100 profit
Max Space: 140 sq ft
Profit Goal: > $1,600
Diagnosis: No Solution
Incompatible Realities
The current constraints create two regions that never overlap in the positive quadrant.
SPACE ≤ 140
PROFIT > 1600
"Any amount that fits won't make enough money. Any amount that makes enough money won't fit."
NO FEASIBLE REGION
Mission: Fix the Plan
1. Form Teams
You are now Strategic Consultants. Work in teams of 3-4.
2. Pivot Strategy
Propose ONE major change to "save" the business:
- Upgrade the truck size?
- Lower the profit goal?
- Renegotiate prices?
3. Prove It
Graph your NEW system. Show the new Feasible Region . Find one valid coordinate.
20:00 MINUTES
PREPARE TO PRESENT
Consultant Mission Brief CONSULTANT MISSION BRIEF
Strategic Operations Unit: Furniture Logistics
CONFIDENTIAL
CASE ID: #SYS-INEQ-402
Consulting Team Name
Analysts (Names)
The Client Case: Warehouse Oversaturation
A regional furniture store is failing to fulfill shipments. Currently, they sell Sofas ($200 profit, 21 sq ft) and Recliners ($100 profit, 9 sq ft) . Their shipment truck has a 140 sq ft limit . Management demands a profit of more than $1,600 per load.
Phase 1: Diagnosis of "No Solution"
1. Write the current System of Inequalities:
SPACE
PROFIT
Mathematical Post-Mortem:
Why is this current system impossible? Look at the intercepts of the two lines. Describe why they will never meet in a way that allows for shipping whole pieces of furniture.
Phase 2: The Business Pivot (Your Strategy)
Management has authorized you to change ONE variable to make the business plan feasible. Choose your strategy:
Option A
Upgrade truck to a larger square footage.
Option B
Lower the profit goal to a realistic number.
Option C
Increase sales price of Sofas or Recliners.
Your New System:
Justification (Calculations):
Explain why your specific number choice (e.g., 200 sq ft or $1200 profit) will work mathematically.
Phase 3: The Valid Solution
After graphing your new system on the official Grid Sheet, identify one specific shipment combination that satisfies your new business plan.
Number of Sofas (x)
Number of Recliners (y)
Grid Masters Graph Paper GRID MASTERS
Strategic Analysis Grid: Project Furniture
AXIS: [X = SOFAS] [Y = RECLINERS]
Number of Sofas (x)
Number of Recliners (y)
0
5
10
15
20
0
5
10
15
20
Intercept Calc
LINE 1: SPACE
x-intercept:
y-intercept:
LINE 2: PROFIT
x-intercept:
y-intercept:
Proposed Pivot
Check your team's strategy:
Upgrade Truck Space
Lower Profit Goal
Increase Pricing
Line Guide
Solid Line (≤ or ≥)
Dashed Line (< or >)
1
Plot the intercepts for BOTH lines. Connect with the correct line type (Solid vs Dashed).
2
Shade the region for each inequality based on your test points. Look for the overlap.
3
Identify your Feasible Region . Choose one (x,y) point as your final recommendation.
Turnaround Teacher Guide TURNAROUND TEACHER GUIDE
Case Study: Furniture Logistics | Algebra 2
TEACHER ONLY
Instructional Objective
Students will analyze a system of linear inequalities that results in a "no solution" scenario. They will apply critical thinking to adjust constraints (intercepts and slopes) to create a feasible region, then identify specific coordinate solutions within that region.
Pacing
Warm-up 5 min
Video View 15 min
Consulting Activity 20 min
Presentations 5 min
PHASE 1: THE FAILED PLAN
The Original System
21x + 9y ≤ 140 (Space)
200x + 100y > 1600 (Profit)
Note: x = sofas, y = recliners. Quadrant I only (x ≥ 0, y ≥ 0).
Graphing Intercepts (Failure Points)
SPACE LINE
x-int: 6.67
y-int: 15.56
PROFIT LINE
x-int: 8.00
y-int: 16.00
Critical Insight: Since both profit intercepts are GREATER than the space intercepts, and the shading for profit goes AWAY from the origin, there is no overlap in Quadrant I.
PHASE 2: PIVOT OPTIONS
Pivot A: Larger Truck
Students increase the constant 140. To work, it must be at least 170.
21x + 9y ≤ 170
Pivot B: Lower Goal
Students decrease the constant 1600. To work, it must be below 1300.
200x + 100y > 1300
Pivot C: Value Add
Increase profit per item (e.g., Sofas become $300 profit). Shifts the slope.
300x + 100y > 1600
Calculator Quick-Start (TI-84 / Desmos)
For TI-84 Plus CE
Go to [Y=].
Move cursor to the left of Y1 . Press [ENTER] to change the line style to "Shade Above" or "Shade Below".
For "No Solution" visual, use [WINDOW]: Xmin: 0, Xmax: 20, Ymin: 0, Ymax: 20.
Must solve for y first: \( y \leq \frac{140 - 21x}{9} \)
For Desmos / Mobile
Type equations exactly as written (Standard Form): 21x + 9y <= 140.
Use the Settings icon to limit to Quadrant I: {x >= 0}{y >= 0}.
Change colors to show overlap (e.g., Blue + Yellow = Green overlap).
