"Mastering Word Problems: Solving One-Step Linear Inequalities"

Solving Word Problems Using One-Step Linear Inequalities

Word problems are a common type of problem that involves real-world scenarios and requires mathematical skills to solve. One-step linear inequalities are a fundamental concept in algebra that can help us solve these problems efficiently. In this article, we will explore how to solve word problems using one-step linear inequalities.

Understanding One-Step Linear Inequalities

Before we dive into solving word problems, let's first understand what one-step linear inequalities are. A one-step linear inequality is an inequality that contains a single variable and has only one mathematical operation (addition or subtraction) applied to it. The general form of a one-step linear inequality is:

ax + b <= c

where 'a' and 'c' are constants, and 'b' is a variable. The inequality symbol (<=) indicates that the expression on the left side of the symbol is less than or equal to the expression on the right side.

Translating Word Problems into Mathematical Expressions

To solve word problems using one-step linear inequalities, the first step is to translate the problem into a mathematical expression. This involves identifying the key information in the problem, assigning variables to represent unknown quantities, and writing an equation or inequality that represents the relationship between these quantities.

For example, consider the following word problem:

Sara is older than her brother by 3 years. If Sara is 15 years old, how old is her brother?

To translate this problem into a mathematical expression, we can assign the variable 'b' to represent the brother's age. The problem states that Sara is older than her brother by 3 years, so we can write the following inequality:

15 - b <= 3

Solving One-Step Linear Inequalities

Once we have translated the word problem into a mathematical expression, the next step is to solve the one-step linear inequality. The process for solving one-step linear inequalities is similar to the process for solving one-step linear equations. Here are the steps to solve a one-step linear inequality:

• Isolate the variable term on one side of the inequality by adding or subtracting the same number from both sides of the inequality.
• If you subtracted a number, reverse the inequality sign. If you added a number, the inequality sign remains the same.
• Simplify the expression to get the variable by itself.

Let's apply these steps to the inequality we derived from the word problem:

15 - b <= 3

Subtract 15 from both sides of the inequality:

b <= -12

Since there is no solution to this inequality (no age can be less than or equal to -12), we must have made a mistake in our translation. Let's re-evaluate the problem and try again.

Re-evaluating the Problem

Upon re-evaluating the problem, we realize that the inequality should be:

b + 3 <= 15

Now, let's apply the steps to solve this inequality:

• Subtract 3 from both sides of the inequality:
• b <= 12
• This inequality has a solution. The brother's age can be any number less than or equal to 12.

Therefore, the brother is 12 years old or younger.

Checking the Solution

It's always a good idea to check our solution by substituting it back into the original inequality. In this case, if the brother is 12 years old, the inequality becomes:

15 - 12 <= 3

Which simplifies to:

3 <= 3

This is a true statement, so our solution is correct.

Practice Problems

Now that you know how to solve word problems using one-step linear inequalities, try your hand at the following practice problems:

Remember to translate each problem into a mathematical expression, solve the one-step linear inequality, and check your solution. With practice, you'll become more proficient at solving word problems using one-step linear inequalities.