inequality word problems worksheet

Guri Bee

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23-10-2024

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Tackling Inequality Word Problems: A Step-by-Step Guide

Inequality word problems can be tricky, but with the right approach, they become manageable. This article breaks down how to tackle these problems, using examples and explanations from helpful resources found on GitHub.

Understanding the Basics

Before diving into word problems, let's review the fundamental concepts of inequalities:

• Inequalities are mathematical statements comparing two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Solving inequalities involves finding the range of values that satisfy the given inequality.

Let's break down a common word problem type:

Example: "Sarah wants to buy a new phone that costs $300. She has already saved $150. How much more money does she need to save each week for the next 5 weeks to buy the phone?"

Step 1: Define the variable. Let 'x' represent the amount Sarah needs to save each week.

Step 2: Set up the inequality. The total amount Sarah needs to save is $300 - $150 = $150. This needs to be less than or equal to the amount she saves each week multiplied by the number of weeks. This translates to: 5x ≥ 150

Step 3: Solve the inequality. Divide both sides by 5: x ≥ 30

Step 4: Interpret the solution. Sarah needs to save at least $30 each week for the next 5 weeks to buy the phone.

Key takeaway: By breaking down the problem into smaller steps, defining variables, and translating the words into an inequality, we can solve for the unknown.

Advanced Word Problems

Let's explore a more complex example:

Example: "A company produces two types of widgets: A and B. Widget A takes 3 hours to produce, and widget B takes 2 hours. The company has a maximum of 48 hours of production time available per day. They want to produce at least twice as many widgets of type B as type A. How many widgets of each type can they produce?"

Solution:

1. Define variables: Let 'a' represent the number of widgets of type A and 'b' represent the number of widgets of type B.

2. Set up inequalities:

   • Production time: 3a + 2b ≤ 48
   • Ratio of widgets: b ≥ 2a

3. Solve the system of inequalities: This requires graphing the inequalities and finding the feasible region, which represents all possible combinations of 'a' and 'b' that satisfy both conditions.

Example:

// Python code to plot the feasible region, taken from GitHub repository: https://github.com/david-guerrero/linear-programming/blob/master/Linear-Programming.ipynb

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(0, 20, 100)
y1 = (48 - 3*x)/2
y2 = 2*x

plt.plot(x, y1, label='3a + 2b ≤ 48')
plt.plot(x, y2, label='b ≥ 2a')
plt.fill_between(x, y1, y2, where=y1>=y2, alpha=0.3)
plt.xlabel('Number of widgets of type A (a)')
plt.ylabel('Number of widgets of type B (b)')
plt.legend()
plt.title('Feasible region for widget production')
plt.show()

Key takeaway: These problems often involve multiple inequalities and require graphical analysis to determine the feasible region.

Additional Resources on GitHub:

• "Linear-Programming.ipynb" by david-guerrero: Provides a comprehensive guide on using Python to solve linear programming problems, including examples of inequality word problems. https://github.com/david-guerrero/linear-programming/blob/master/Linear-Programming.ipynb
• "inequality.py" by cody-ray: A Python script illustrating how to solve inequalities using symbolic math tools. https://github.com/cody-ray/inequality/blob/main/inequality.py

By exploring these resources, you can gain further insights into solving various inequality word problems and master this crucial mathematical concept.