linear inequality word problems

Guri Bee

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19-10-2024

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Solving Real-World Problems with Linear Inequalities: A Step-by-Step Guide

Linear inequalities are mathematical expressions that involve variables, constants, and inequality signs like <, >, ≤, and ≥. They are used to represent relationships where one quantity is greater than, less than, greater than or equal to, or less than or equal to another quantity. These inequalities are incredibly useful in solving real-world problems, helping us understand limitations, optimize resources, and make informed decisions.

Let's explore how to approach linear inequality word problems with a practical example:

Scenario:

Imagine you're planning a birthday party for your friend. You have a budget of $150 for decorations. Balloons cost $5 per pack, and streamers cost $2 per pack. You want to buy at least 5 packs of balloons but no more than 15 packs of streamers. How many packs of each can you buy?

Step 1: Define Variables

• Let "b" represent the number of packs of balloons.
• Let "s" represent the number of packs of streamers.

Step 2: Formulate Inequalities

• Budget Constraint: The total cost of balloons and streamers should not exceed $150. This translates to the inequality: 5b + 2s ≤ 150
• Balloon Minimum: You want to buy at least 5 packs of balloons, represented by: b ≥ 5
• Streamer Maximum: You want to buy no more than 15 packs of streamers, represented by: s ≤ 15

Step 3: Solve the System of Inequalities

To find the possible combinations of balloons and streamers, we need to graph the system of inequalities. This will give us a shaded region representing all the possible solutions.

• Graphing 5b + 2s ≤ 150:
  • First, we need to find the intercepts. If b = 0, then s = 75. If s = 0, then b = 30. Plot these points and draw a line connecting them.
  • Since the inequality is "less than or equal to", the shaded region will be below the line.
• Graphing b ≥ 5:
  • Draw a vertical line at b = 5.
  • Since the inequality is "greater than or equal to", the shaded region will be to the right of the line.
• Graphing s ≤ 15:
  • Draw a horizontal line at s = 15.
  • Since the inequality is "less than or equal to", the shaded region will be below the line.

The overlapping shaded area represents all the possible combinations of balloons and streamers that satisfy all the conditions.

Step 4: Interpret the Solution

Any point within the shaded region is a valid solution. For example, you could buy 10 packs of balloons and 10 packs of streamers. This combination would satisfy all the conditions:

• Budget: 5(10) + 2(10) = 70, which is less than $150.
• Balloon Minimum: 10 packs of balloons is greater than 5.
• Streamer Maximum: 10 packs of streamers is less than 15.

Additional Considerations:

• You could use the graph to explore different scenarios, such as finding the combination that maximizes the number of balloons or streamers while staying within budget.
• You might consider rounding the number of balloons and streamers to whole numbers since you can't purchase fractional packs.

Key Takeaways:

• Linear inequalities are powerful tools for modeling real-world situations involving constraints and limits.
• By defining variables, formulating inequalities, solving them graphically or algebraically, and interpreting the solutions, we can make informed decisions.
• Real-world problems often involve multiple inequalities that must be considered simultaneously.

By learning to solve linear inequality word problems, you gain a valuable skill for problem-solving in various areas of life, from budgeting and planning to understanding financial data and even making informed choices in your daily life.

References:

This article utilizes a common example scenario often found in algebra textbooks. You can find more resources and practice problems on linear inequalities by searching online or consulting your algebra textbook.