Q 322 - Intermediate Algebra

Question

Jocelyn is pregnant and so she needs to eat at least $500$ more calories a day than usual. When buying groceries one day with a budget of \($15$ for the extra food, she buys bananas that have $90$ calories each and chocolate granola bars that have $150$ calories each. The bananas cost \)$0.35$ each and the granola bars cost $$2.50$ each.

ⓐ Write a system of inequalities to model this situation. 

ⓑ Graph the system. 

ⓒ Could she buy $5$ bananas and $6$ granola bars?

ⓓ Could she buy $3$ bananas and $4$ granola bars? 

Step-by-Step Solution

Answer

Part a. A system of inequalities to model this situation is $\left\{\begin{array}{l}90x+150y\ge 500\\ 0.35x+2.50y\le 15\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part b. The graph of the system is

Part c. She cannot buy $5$ bananas and $6$ granola bars

Part d. She can buy $3$ bananas and $4$ granola bars

Part (a) Step 1. Form the system of inequalities

Let $x$ represents the number of bananas bought and $y$ represents the number of granola bars bought.

The calories in one banana is $90$ and the calories in one granola bar is $150$, she need at least $500$ calories a day, so an inequality can be written as

$90x+150y\ge 500$

The price of one banana is $$0.35$and the price of one granola bar is $$2.5$. She can spend a maximum of $$15$, so another inequality can be

$0.35x+2.50y\le 15$

Now number of bananas and garnola bars will always be a non-negative quantity, so the system of inequalities is 

$\left\{\begin{array}{l}90x+150y\ge 500\\ 0.35x+2.50y\le 15\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 1. Graph the first inequality

Draw a solid line $90x+150y=500$ and the point $(0,0)$ does not satisfy the inequality $90x+150y\ge 500$. So shade the region black that does not contain the point $(0,0)$ to get the graph.

Part (b) Step 2. Graph the second inequality on the same coordinate plane

Draw a solid line $0.35x+2.50y=15$ and the point $(0,0)$ satisfy the inequality $0.35x+2.50y\le 15$. So shade the region red that contains the point $(0,0)$ to get the graph.

The common shaded region is the solution of the system of inequality. 

Part (c) Step 1. Check the point on the graph

$5$bananas and $6$granola bars represent the point $(5,6)$ on the coordinate plane, so plotting the point we get

The point does not lie on the common shaded region, so she cannot buy $5$ bananas and $6$ granola bars.

Part (d) Step 1. Check the point on the graph

$3$ bananas and $4$ granola bars represent the point $(3,4)$ on the coordinate plane, so plotting the point we get

The point lies on the common shaded region, so she can buy bananas and $4$ granola bars.