Question

eula needs to buy binders that cost $4 each and notebooks that cost $2 each. she has $20. the graph of the inequality $4x + 2y \leq 20$, which represents the situation, is shown.

what is the greatest number of binders eula can buy?

what is the greatest number of notebooks eula can buy?

if eula buys 7 notebooks, what is the greatest number of binders she can buy?

(graph: x - axis is number of binders, y - axis is number of notebooks, the shaded region is under the line from (0,9) to (5,0))

Explanation:

First Question: Greatest number of binders

Step1: Set notebooks to 0

To find the greatest number of binders, we assume Eula buys 0 notebooks (so \( y = 0 \)) in the inequality \( 4x + 2y \leq 20 \).

Step2: Solve for x

Substitute \( y = 0 \) into the inequality: \( 4x + 2(0) \leq 20 \), which simplifies to \( 4x \leq 20 \). Then divide both sides by 4: \( x \leq \frac{20}{4} = 5 \).

Step1: Set binders to 0

To find the greatest number of notebooks, we assume Eula buys 0 binders (so \( x = 0 \)) in the inequality \( 4x + 2y \leq 20 \).

Step2: Solve for y

Substitute \( x = 0 \) into the inequality: \( 4(0) + 2y \leq 20 \), which simplifies to \( 2y \leq 20 \). Then divide both sides by 2: \( y \leq \frac{20}{2} = 10 \).

Step1: Substitute y = 7

We know \( y = 7 \) (7 notebooks), substitute into the inequality \( 4x + 2y \leq 20 \): \( 4x + 2(7) \leq 20 \).

Step2: Simplify and solve for x

Simplify the left side: \( 4x + 14 \leq 20 \). Subtract 14 from both sides: \( 4x \leq 20 - 14 = 6 \). Then divide by 4: \( x \leq \frac{6}{4} = 1.5 \). Since the number of binders must be a non - negative integer, the greatest integer less than or equal to 1.5 is 1.

Answer:

5

Second Question: Greatest number of notebooks