One Step Inequalities Word Problems Worksheet

One-Step Inequalities Word Problems Worksheet: A Comprehensive Guide

Solving word problems is a crucial skill in mathematics, and understanding inequalities is essential for tackling real-world scenarios. This article delves into one-step inequalities word problems, providing a comprehensive guide with examples, strategies, and practice exercises to boost your problem-solving skills. We'll move from basic understanding to advanced techniques, ensuring you master this vital mathematical concept.

Understanding One-Step Inequalities

Before diving into word problems, let's solidify our understanding of one-step inequalities. Inequalities, unlike equations, show a relationship where one side is greater than, less than, greater than or equal to, or less than or equal to the other side. This is represented by symbols:

• > Greater than
• < Less than
• ≥ Greater than or equal to
• ≤ Less than or equal to

A one-step inequality involves only one operation (addition, subtraction, multiplication, or division) needed to isolate the variable. Solving involves performing the inverse operation on both sides of the inequality, remembering to flip the inequality sign if multiplying or dividing by a negative number.

Example: Solving a Basic One-Step Inequality

Let's solve the inequality: x + 5 > 10

1. Identify the operation: Addition (5 is added to x)
2. Perform the inverse operation: Subtract 5 from both sides.
3. x + 5 - 5 > 10 - 5
4. Simplify: x > 5

The solution is all values of x greater than 5.

Tackling One-Step Inequalities Word Problems

Word problems present inequalities in real-world contexts. The key is to translate the written description into a mathematical inequality and then solve it. Let's break down the process with some examples:

Example 1: The Savings Goal

Problem: Sarah wants to save at least $150 for a new bicycle. She has already saved $75. How much more money does she need to save?

Solution:

1. Define the variable: Let 'x' represent the amount of money Sarah still needs to save.
2. Translate into an inequality: The total savings ($75 + x) must be greater than or equal to $150. This translates to: 75 + x ≥ 150
3. Solve the inequality: Subtract 75 from both sides: x ≥ 75
4. Interpret the solution: Sarah needs to save at least $75 more.

Example 2: The Temperature Limit

Problem: The temperature inside a refrigerator should be less than 40°F. The current temperature is 45°F. By how much must the temperature be lowered?

Solution:

1. Define the variable: Let 'x' represent the amount the temperature must be lowered.
2. Translate into an inequality: The final temperature (45 - x) must be less than 40°F. This gives us: 45 - x < 40
3. Solve the inequality: Add 'x' and subtract 40 from both sides: 5 < x This can be rewritten as x > 5
4. Interpret the solution: The temperature must be lowered by more than 5°F.

Example 3: The Weight Limit

Problem: A truck can carry a maximum weight of 2000 lbs. It is already carrying 1250 lbs. How many more pounds can it carry?

Solution:

1. Define the variable: Let 'x' represent the additional weight the truck can carry.
2. Translate into an inequality: The total weight (1250 + x) must be less than or equal to 2000 lbs. This translates to: 1250 + x ≤ 2000
3. Solve the inequality: Subtract 1250 from both sides: x ≤ 750
4. Interpret the solution: The truck can carry at most 750 more pounds.

Example 4: The Speed Limit

Problem: The speed limit on a highway is 65 mph. A car is traveling at a speed of 's' mph. Write an inequality to represent the situation where the car is exceeding the speed limit.

Solution:

This problem doesn't require solving, but rather writing the inequality. The car's speed (s) must be greater than 65 mph to exceed the speed limit. The inequality is: s > 65

Advanced Techniques and Considerations

While one-step inequalities are relatively straightforward, several advanced techniques and considerations can enhance your problem-solving capabilities:

1. Compound Inequalities

Some word problems involve compound inequalities, which combine two or more inequalities. For example: "The temperature must be between 60°F and 80°F." This translates to 60 < T < 80, where T represents the temperature.

2. Contextual Understanding

Always carefully read and understand the context of the problem. Pay attention to keywords like "at least," "at most," "more than," "less than," "no more than," "no less than," etc., to correctly represent the inequality.

3. Checking Your Solution

After solving an inequality, always check your answer by substituting a value within the solution set back into the original inequality. It should make the inequality true.

4. Graphing the Solution

Graphing the solution set on a number line provides a visual representation of the inequality's solution. This helps in understanding the range of values that satisfy the inequality.

Practice Problems: One-Step Inequalities Word Problems Worksheet

Here are some practice problems to test your understanding:

1. A student needs to score at least 80% on the final exam to pass the course. If the student scored 72% on the midterm, how many percentage points must they improve on the final exam to pass?

2. A baker can bake a maximum of 50 cakes per day. If they have already baked 25 cakes, how many more cakes can they bake today?

3. The maximum capacity of an elevator is 1000 lbs. If 5 people weighing an average of 180 lbs are already in the elevator, how much additional weight can the elevator carry?

4. A taxi charges $3 for the first mile and $2 for each additional mile. If a passenger has a maximum of $20 to spend, how many miles can they travel?

5. A phone plan allows for up to 500 minutes of talk time per month. If a user has already used 280 minutes, how many more minutes can they use this month?

These problems provide a good starting point. Remember to meticulously translate the word problems into mathematical inequalities and then systematically solve them, always double-checking your work. Consistent practice will significantly improve your skill in solving one-step inequalities word problems, making you more confident in handling real-world mathematical situations.