Question 1191395: Translate to a system of inequalities and solve.
Jocelyn is pregnant and needs to eat at least 400 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.
(a)
Write a system of inequalities to model this situation. (Let b represent the bananas bought and let g represent the number of granola bars bought. Enter your answers as a comma-separated list of inequalities.)
(b)
Graph the system.
(c)
Could she buy 6 bananas and 6 granola bars?
(d)
Could she buy 3 bananas and 3 granola bars?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **(a) System of Inequalities**
Let 'b' represent the number of bananas and 'g' represent the number of granola bars.
* **Calorie Inequality:** Jocelyn needs at least 400 extra calories.
90b + 150g ≥ 400
* **Budget Inequality:** Jocelyn has a budget of $15.
0.35b + 2.50g ≤ 15
* **Non-Negative Inequalities:** Jocelyn can't buy a negative number of bananas or granola bars.
b ≥ 0
g ≥ 0
Therefore, the system of inequalities is: **90b + 150g ≥ 400, 0.35b + 2.50g ≤ 15, b ≥ 0, g ≥ 0**
**(b) Graphing the System**
I can't draw a graph here, but I'll describe how to graph it:
1. **Convert to Slope-Intercept Form (for easier graphing):**
* Calorie Inequality: 150g ≥ -90b + 400 => g ≥ (-3/5)b + (8/3)
* Budget Inequality: 2.50g ≤ -0.35b + 15 => g ≤ (-7/50)b + 6
2. **Plot the Lines:**
* For g ≥ (-3/5)b + (8/3), plot the line g = (-3/5)b + (8/3). Since it's "greater than or equal to," shade the area *above* this line.
* For g ≤ (-7/50)b + 6, plot the line g = (-7/50)b + 6. Since it's "less than or equal to," shade the area *below* this line.
* Also, remember b ≥ 0 and g ≥ 0, which means you're only considering the first quadrant (positive b and g values).
3. **Solution Region:** The solution region is where all the shaded areas overlap. This represents the possible combinations of bananas and granola bars that meet Jocelyn's requirements.
**(c) 6 Bananas and 6 Granola Bars?**
* Calories: 90(6) + 150(6) = 540 + 900 = 1440 calories (meets the calorie need)
* Cost: 0.35(6) + 2.50(6) = 2.10 + 15 = $17.10 (exceeds the budget)
No, she could not buy 6 bananas and 6 granola bars because it exceeds her budget.
**(d) 3 Bananas and 3 Granola Bars?**
* Calories: 90(3) + 150(3) = 270 + 450 = 720 calories (meets the calorie need)
* Cost: 0.35(3) + 2.50(3) = 1.05 + 7.50 = $8.55 (within the budget)
Yes, she could buy 3 bananas and 3 granola bars.
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