Two-Step Inequality Word Problems Worksheet With Answers

Understanding and Solving Two-Step Inequality Word Problems

Word problems involving inequalities can often be challenging due to the need to translate words into mathematical expressions and then solve them. Two-step inequality word problems, in particular, require you to perform two operations to solve the inequality, making them slightly more complex than one-step inequalities. This article will guide you through understanding, translating, and solving two-step inequality word problems with a focus on practical examples and solutions.

Translating Word Problems into Mathematical Expressions

To solve two-step inequality word problems, the first step is to translate the words into a mathematical expression. This involves identifying the key elements:

• Variables: The unknown quantity you’re trying to solve for.
• Constants: Numbers that don’t change.
• Operations: Addition, subtraction, multiplication, or division.
• Inequality Sign: Determines the relationship between the expressions.

Consider the following example:

Example 1: Sarah has been saving money for a new bike and has 120 in her savings account. She wants to buy a bike that costs 180. If she earns $12 per hour by walking dogs, how many hours does she need to work to have enough money to buy the bike?

Let’s translate this into a mathematical expression:

[ 120 + 12x > 180 ]

Where (x) is the number of hours Sarah needs to work.

Solving Two-Step Inequality Word Problems

Solving these inequalities involves performing two operations to isolate the variable. Here’s how to solve the inequality from Example 1:

[ 120 + 12x > 180 ]

1. First Step: Subtract 120 from both sides of the inequality to get the term with the variable by itself on one side.

[ 12x > 180 - 120 ] [ 12x > 60 ]

1. Second Step: Divide both sides by 12 to solve for (x).

[ x > \frac{60}{12} ] [ x > 5 ]

So, Sarah needs to work more than 5 hours to earn enough money to buy the bike.

Another Example and Solution

Example 2: A bookshelf has 5 shelves, and each shelf can hold 8 books. If the bookshelf is currently empty, how many books can be placed on it in total? If the owner already has 20 books and wants to buy more, how many more books can he buy if he wants the total number of books (the ones he already has plus the new ones) to be less than the capacity of the bookshelf?

First, calculate the total capacity of the bookshelf:

[ 5 \times 8 = 40 \text{ books} ]

The inequality representing the situation where the owner wants the total number of books to be less than the capacity of the bookshelf is:

[ 20 + x < 40 ]

Where (x) is the number of additional books the owner can buy.

1. First Step: Subtract 20 from both sides.

[ x < 40 - 20 ] [ x < 20 ]

1. Second Step: There’s no need for a second step in this case, as (x) is already isolated.

So, the owner can buy less than 20 more books.

Important Notes

• When solving inequalities, the direction of the inequality sign can change depending on the operation performed. Specifically, when dividing or multiplying by a negative number, the inequality sign is flipped.
• Always read the problem carefully to ensure you understand what is being asked. This will help you set up the correct inequality.

💡 Note: Practice is key to becoming proficient in solving two-step inequality word problems. Try to create your own examples or find worksheets to practice your skills.

The ability to translate word problems into mathematical expressions and solve them is a valuable skill in mathematics and real-life problem-solving. By following the steps outlined in this guide and practicing with different types of problems, you can become proficient in solving two-step inequality word problems.

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Related Terms:

• Solving inequalities word problems
• One-step inequalities word problems worksheet