**Greatest number of binders:**- The inequality is \(4x + 2y \leq 20\).
- To find the maximum number of binders (\(x\)), set \(y = 0\):

\[4x \leq 20 \implies x \leq 5\] - Therefore, the greatest number of binders Eula can buy is 5.

**Greatest number of notebooks:**- To find the maximum number of notebooks (\(y\)), set \(x = 0\):

\[2y \leq 20 \implies y \leq 10\] - Therefore, the greatest number of notebooks Eula can buy is 10.

**If Eula buys 7 notebooks:**- Substitute \(y = 7\) into the inequality:

\[4x + 2( 7) \leq 20 \implies 4x + 14 \leq 20 \implies 4x \leq 6 \implies x \leq 1.5\] - Since \(x\) must be an integer, the greatest number of binders she can buy is 1.

#### Supplemental Knowledge

Linear inequalities involve expressions that use inequality symbols (such as â‰¤, <, â‰Ą, >) instead of an equals sign. They describe a range of possible solutions rather than a single solution. When graphing linear inequalities, the solution set is represented by a shaded region on one side of the boundary line.

#### Key Concepts:

**Graphing Linear Inequalities:**- The inequality \(4x + 2y \leq 20\) can be graphed by first plotting the boundary line \(4x + 2y = 20\).
- To plot this line, find the intercepts:
- Set \(x = 0\): \(2y = 20 \Rightarrow y = 10\)
- Set \(y = 0\): \(4x = 20 \Rightarrow x = 5\)

- Draw the line through these points and shade the region that satisfies the inequality.

**Interpreting Solutions:**- The feasible region (shaded area) represents all possible combinations of binders (\(x\)) and notebooks (\(y\)) that Eula can buy within her budget.
- Integer solutions within this region represent realistic purchasing options since Eula cannot buy fractional binders or notebooks.

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#### Theory in Practice

- Consider budgeting scenarios where limited resources must be distributed among various categories. For instance, when spending on groceries and household essentials with limited money available for spending, understanding linear inequalities to help make informed decisions is paramount to making accurate estimates and budgets.

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