Inequalities and Constraints

Slope and Units

  • Matching units to make slope calculations more intuitive.
  • Mathematically correct representation: down 5 units over 10 units.

Practical Constraints

  • Ice cream scoops: The cap is around 35 scoops because exceeding this might lead to losses.
  • Negative slopes: Represented by using more small scoops, leading to fewer large scoops, especially considering cheese scoops use double the amount.

Adjusting Slope Representation

  • Transforming negative slopes: Example: -\frac{3}{5} can be scaled to -\frac{15}{25}.

Graphing Inequalities

  • Graphing regions: Shading below the blue line for 'less than' and above the red line for 'greater than'.
  • Sweet spot: The area where the colors merge (purple) represents the solution that satisfies both inequalities.

Business Implications

  • Real-world application: Balancing the number of small and large ice cream scoops based on a total of 70 scoops and a target profit of at least $20.
  • If you sell 20 small scoops, you need to sell a certain number of larger ones to make money.
  • Selling only 10 large scoops results in a loss.

Setting Up a Summer Party: Hamburgers and Chicken

  • Cost: Hamburgers at $2 per pound, chicken at $3 per pound.
  • Budget: No more than $30 to spend.
  • Minimum Purchase: At least 3 pounds of hamburgers.
  • What-if Scenario: If only hamburgers are bought, buying at least three pounds

Defining Variables

  • Hamburgers: Independent variable (x-axis).
  • Chicken: Dependent variable (y-axis), its quantity depends on the hamburger purchase.

Formulating Inequalities

  • Money Inequality: 2x + 3y \le 30, where x is pounds of hamburgers and y is pounds of chicken.
  • Pound Inequality: x \ge 3, representing the minimum purchase of hamburgers.

Solving and Simplifying Inequalities

  • Transforming the money inequality: Solving for y to get y \le -\frac{2}{3}x + 10.
  • Determining scale: Assessing the simplified inequality to decide on appropriate units for the graph (e.g., ones).

Graphing the Inequalities

  • First Inequality: Starting at 10 on the y-axis, with a slope of down 2 over 3.
  • Shading: Below the line for 'less than'.
  • Second Inequality: A vertical line at x = 3, shading to the right for 'greater than'.

Analyzing the Graph

  • Maximum Hamburger Meat: Purchasing only hamburger meat, the maximum is 15 pounds.
  • Testing the solution area: Using test points to confirm the correct shaded region.

Interpreting the Results

  • The overlapping region indicates feasible solutions.
  • Example 1: Buying 10 pounds of chicken requires no hamburger meat, which is outside the solution area.