Linear Inequalities, Budget Equations, and Graph Interpretation

Boundary Line Types:
- Dashed Line: Represents strict inequalities containing less than ( $<$ ) or greater than ( $>$ ) symbols. Points located directly on a dashed line are not included in the solution set.
- Solid Line: Represents inclusive inequalities containing less than or equal to ( $\le$ ) or greater than or equal to ( $\ge$ ) symbols. Points located directly on a solid line are included in the solution set.
Shading Directions:
- Shaded Below / Bottom: Represents values that are less than ( $<$ or $\le$ ) the boundary line.
- Shaded Above / Top: Represents values that are greater than ( $>$ or $\ge$ ) the boundary line.
Matching Inequalities to Base Graphs (where $\text{y-intercept} = 0$ and $\text{slope} = 1$ ):
- Graph 1: Dashed line shaded below $\rightarrow$ $y < x$ (Inequality #3).
- Graph 2: Dashed line shaded above $\rightarrow$ $y > x$ (Inequality #1).
- Graph 3: Solid line shaded above $\rightarrow$ $y \ge x$ (Inequality #2).
- Graph 4: Solid line shaded below $\rightarrow$ $y \le x$ (Inequality #4).

Evaluating Ordered Pair $(2, 2)$ :
- The coordinate $(2, 2)$ lies directly on the line $y = x$
- Graph B ( $y \ge x$ ) and Graph D ( $y \le x$ ) contain solid lines. Because the boundary line is solid, points directly on the line are included in the solution set. Therefore, $(2, 2)$ is a solution for Graphs B and D.
- Graph C ( $y < x$ ) contains a dashed line. Points directly on a dashed boundary line are excluded from the solution set. Thus, $(2, 2)$ is not a solution for Graph C.

Scenario Parameters:
- Grass sod cost ( $x$ ): $2\,\text{dollars per square foot}$ .
- Flower bed cost ( $y$ ): $12\,\text{dollars per square foot}$ .
- Total budget limit: $3000\,\text{dollars}$ .
Standard Form Linear Equation:
- Using the exact entire budget requires an equality equation ( $=$ , containing an equal sign): $2x + 12y = 3000$
Determining Intercepts for Graphing:
- $x$ -intercept (Value of $x$ when $y = 0$ ): $2x + 12(0) = 3000$ $2x = 3000$ $x = 1500$ Coordinates: $(1500, 0)$
- $y$ -intercept (Value of $y$ when $x = 0$ ): $2(0) + 12y = 3000$ $12y = 3000$ $y = 250$ Coordinates: $(0, 250)$
Graphing the Line:
- Plot the two intercept points $(1500, 0)$ and $(0, 250)$ on the coordinate plane.
- Draw a straight line connecting these two points. Alternatively, convert the equation into slope-intercept form $y = -\frac{1}{6}x + 250$ .

Concept of Solutions:
- Any coordinate $(x, y)$ lying directly on the line $2x + 12y = 3000$ represents a valid combination of sod and flower beds that totals exactly $3000\,\text{dollars}$ .
Solution Proof Example:
- Point $(1500, 0)$ : $2(1500) + 12(0) = 3000$ $3000 + 0 = 3000$ $3000 = 3000$ This is a valid solution.
Non-Solution Proof Example:
- Point $(2000, 400)$ : $2(2000) + 12(400) = 4000 + 4800$ $4000 + 4800 = 8800$ $8800 \neq 3000$ Since $8800\,\text{dollars}$ far exceeds the $3000\,\text{dollars}$ budget, $(2000, 400)$ is a non-solution.

Scenario Parameters:
- Gravel cost ( $x$ ): $3\,\text{dollars per square foot}$ .
- Artificial turf cost ( $y$ ): $15\,\text{dollars per square foot}$ .
- Total budget: $3000\,\text{dollars}$ .
Standard Form Linear Equation: $3x + 15y = 3000$
Contextual Meaning of Coordinates:
- Point $(500, 100)$ : Represents installing $500\,\text{square feet}$ of gravel and $100\,\text{square feet}$ of artificial turf.
- Verification: $3(500) + 15(100) = 1500 + 1500 = 3000\,\text{dollars}$ .
- Extremes on the line:
- $(1000, 0)$ : $1000\,\text{sq ft}$ gravel and $0\,\text{sq ft}$ turf ( $3(1000) = 3000$ ).
- $(0, 200)$ : $0\,\text{sq ft}$ gravel and $200\,\text{sq ft}$ turf ( $15(200) = 3000$ ).

Points Above and to the Right of the Line (Exceeding Budget):
- Point $(600, 200)$ : $3(600) + 15(200) = 1800 + 3000 = 4800$ $4800 > 3000$ Does not meet constraints because $4800\,\text{dollars}$ is over budget.
- Point $(1000, 200)$ : $3(1000) + 15(200) = 3000 + 3000 = 6000$ $6000 > 3000$ Does not meet constraints ( $6000\,\text{dollars}$ exceeds budget).
Points Below and to the Left of the Line (Within Budget):
- Point $(200, 100)$ : $3(200) + 15(100) = 600 + 1500 = 2100$ $2100 \le 3000$ Meets constraints because spending $2100\,\text{dollars}$ is within the $3000\,\text{dollars}$ limit.
- Point $(0, 0)$ : $3(0) + 15(0) = 0$ $0 \le 3000$ Meets constraints (purchasing nothing costs $0\,\text{dollars}$ ).
Out-of-Bounds Check:
- Point $(75, 200)$ : $3(75) + 15(200) = 225 + 3000 = 3225$ $3225 > 3000$ Exceeds budget and lies above the boundary line.

Inequality Formulation:
- To reflect a budget limit where total cost must be less than or equal to $3000\,\text{dollars}$ : $3x + 15y \le 3000$
Shading and Graph Interpretations:
- Boundary Line: Draw a solid line for $3x + 15y = 3000$ because the symbol is $\le$ (includes points that cost exactly $3000\,\text{dollars}$ ).
- Solution Region: Shade the entire area below and to the left of the line.
- Region Meanings:
- Points on the line: Combinations costing exactly $3000\,\text{dollars}$ .
- Points in shaded region below line: Combinations costing less than $3000\,\text{dollars}$ .
- Points in unshaded region above line: Combinations exceeding the $3000\,\text{dollars}$ budget.

Collaborative Task Workflow:
- Divide into groups of 3 students.
- Assign individual tasks:
- Task 2: Saturday Market
- Task 3: Bank Accounts / Discounts
- Task 4: Concert Tickets
- Execution Steps:
1. Complete assigned individual task independently.
2. Peer Instruction: Teach the completed task solution to other group members.
3. Peer Transcription: Record the explanations and solutions provided by group members for the remaining tasks.