1. Greatest Number of Binders:
   • The inequality is \(4x + 2y \leq 20\).
   • Set \(y = 0\) to find the maximum \(x\):
     \[4x + 2( 0) \leq 20\]
     \[4x \leq 20\]
     \[x \leq 5\]
   • Therefore, the greatest number of binders Eula can buy is 5.
2. Greatest Number of Notebooks:
   • Set \(x = 0\) to find the maximum \(y\):
     \[4( 0) + 2y \leq 20\]
     \[2y \leq 20\]
     \[y \leq 10\]
   • Therefore, the greatest number of notebooks Eula can buy is 10.
3. Greatest Number of Binders if Eula Buys 7 Notebooks:
   • Substitute \(y = 7\) into the inequality:
     \[4x + 2( 7) \leq 20\]
     \[4x + 14 \leq 20\]
     \[4x \leq 6\]
     \[x \leq 1.5\]
   • Since \(x\) must be an integer, the greatest number of binders Eula can buy is 1.

Supplemental Knowledge

Linear inequalities are mathematical expressions involving a linear function where the values of the variables are constrained by inequality symbols (≤, <, ≥, >). When graphing linear inequalities, the solution set is represented by a shaded region on one side of the boundary line (which represents the equation part of the inequality).
For example, if you have an inequality like \(4x + 2y \leq 20\):

1. Graph the Boundary Line: First, convert it to an equation \(4x + 2y = 20\) and graph this line.
2. Determine Shading: Choose a test point not on the line (usually (0,0)) to determine which side of the line to shade.
3. Interpret Solutions: Any point in the shaded region satisfies the inequality.

Theory in Practice

1. Consider budgeting for groceries. If you have 20 and need to buy fruits and vegetables where fruits cost $4 per unit and vegetables cost $2 per unit:

• You can use a similar inequality to determine how many units of each you can afford.
• Graphing this helps visualize your options and make informed decisions.
  This approach is practical for various budgeting scenarios, helping manage finances effectively.

Understanding how to solve and graph linear inequalities can significantly enhance your problem-solving skills in real-life situations. At UpStudy, we offer comprehensive resources designed to help you master these essential mathematical concepts.
For detailed guidance on solving algebraic inequalities, explore UpStudy’s Algebra Inequalities Calculator today!