Problem 319 - Elementary Algebra

Question

Jocelyn is pregnant and 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. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could she buy 5 bananas and 6 granola bars? (d) Could she buy 3 bananas and 4 granola bars?

Step-by-Step Solution

Answer

The system of inequalities is: \( 90b + 150g \geq 500 \) and \( 0.35b + 2.50g \leq 15 \). Jocelyn can buy 3 bananas and 4 granola bars, but cannot buy 5 bananas and 6 granola bars.

Step 1: Define the variables

Let \( b \) represent the number of bananas and \( g \) represent the number of granola bars.

Step 2: Write the calorie inequality

Jocelyn needs at least 500 extra calories, so write the inequality for the calories: \[ 90b + 150g \geq 500 \]

Step 3: Write the budget inequality

Jocelyn has a budget of $15, so write the inequality for the budget: \[ 0.35b + 2.50g \leq 15 \]

Step 4: Graph the inequalities

Graph the inequalities on a coordinate plane, where the x-axis represents the number of bananas \( b \) and the y-axis represents the number of granola bars \( g \).

Step 5: Check point \( (5, 6) \)

Substitute \( b = 5 \) and \( g = 6 \) into both inequalities: 1. Calorie Inequality: \[ 90(5) + 150(6) = 450 + 900 = 1350 \geq 500 \] 2. Budget Inequality: \[ 0.35(5) + 2.50(6) = 1.75 + 15 = 16.75 \ot\leq 15 \] Since the second inequality is not satisfied, Jocelyn cannot buy 5 bananas and 6 granola bars.

Step 6: Check point \( (3, 4) \)

Substitute \( b = 3 \) and \( g = 4 \) into both inequalities: 1. Calorie Inequality: \[ 90(3) + 150(4) = 270 + 600 = 870 \geq 500 \] 2. Budget Inequality: \[ 0.35(3) + 2.50(4) = 1.05 + 10 = 11.05 \leq 15 \] Both inequalities are satisfied, so Jocelyn can buy 3 bananas and 4 granola bars.

Key Concepts

inequalities in mathematics

Inequalities are mathematical expressions that show the relationship between two values that are not equal. Rather than using an equal sign such as in equations, inequalities use symbols like \( > \) (greater than), \( < \) (less than), \( \geq \) (greater than or equal to), and \( \leq \) (less than or equal to). For example, Jocelyn’s caloric needs and budget constraints are nicely handled using inequalities. If we represent the number of bananas by \( b \) and granola bars by \( g \), her caloric and budget needs can be written as:

• \( 90b + 150g \geq 500 \) for calories.
• \( 0.35b + 2.50g \leq 15 \) for budget.

These inequalities help us explore multiple possibilities within defined limits, letting us understand which combinations of bananas and granola bars meet her nutritional and financial requirements.

budget constraints

Budget constraints represent the limitation of spending within a specific financial boundary. They are crucial for managing resources effectively. In Jocelyn’s case, her budget constraint equation is represented by the inequality: \[0.35b + 2.50g \leq 15 \] where \( b \) is the number of bananas she can buy at \( \$0.35 \) each, and \( g \) is the number of granola bars she can buy at \( \$2.50 \) each.

• The total cost of bananas and granola bars should not exceed her budget of \( \$15 \).
• This inequality ensures she does not overspend while trying to meet her caloric needs.

By balancing the inequalities, Jocelyn can find combinations of bananas and granola bars that stay within her budget while providing enough calories.

caloric intake equations

Caloric intake equations are used to quantify the amount of energy, measured in calories, obtained from food. In Jocelyn’s scenario, she needs at least 500 additional calories, modeled by the inequality: \[90b + 150g \geq 500 \] where \( 90 \) represents the calories per banana and \( 150 \) represents the calories per granola bar.

• The sum of the calories from bananas and granola bars should meet or exceed 500 calories:
• This ensures Jocelyn achieves her necessary dietary intake.

The system of equations helps manage both the nutritional and financial aspects of her diet, providing clarity on how many food items to buy within her constraints.

graphing inequalities

Graphing inequalities provides a visual way to represent solutions that satisfy a system of inequalities. For Jocelyn’s calorie and budget needs, each inequality forms a region on a graph. To graph her constraints:

• Plot \( 90b + 150g \geq 500 \) and \( 0.35b + 2.50g \leq 15 \) on the same number plane.
• The x-axis represents the number of bananas \( b \) and the y-axis represents the number of granola bars \( g \).
• The feasible solutions are the points lying in the intersection of the shaded regions formed by these inequalities.

This visualization helps to understand which combinations of \( b \) and \( g \) satisfy both constraints, making it easier to identify the practical choices for Jocelyn within her defined limits.