Solving One-Step Linear Inequalities: A Comprehensive Guide

Unlock the power of one-step linear inequalities with our easy-to-follow guide. Learn essential techniques, practice with real-world examples, and boost your algebra skills today.

Solve. $7x \le 63$

Step 1: Understand the Inequality

When solving one-step linear inequalities, it's important to understand that the process is similar to solving linear equations. The main difference is the inequality sign (, , <, >) instead of an equal sign (=). In this case, we have the inequality $7x \le 63$. Our goal is to isolate the variable $x$ on one side of the inequality.

Step 2: Treat the Inequality Like an Equation

To simplify the process, treat the inequality as if it were an equation. This means you can perform the same operations on both sides of the inequality as you would with an equation. For instance, if you had $7x = 63$, you would solve for $x$ by dividing both sides by 7. The same approach applies here.

Step 3: Divide Both Sides by 7

To isolate $x$, divide both sides of the inequality by 7. This operation will help you get $x$ by itself on one side of the inequality. Here's how it looks:

$7x \le 63$

Divide both sides by 7:

$\frac{7x}{7} \le \frac{63}{7}$

When you divide 7 by 7 on the left side, it cancels out, leaving you with $x$. On the right side, 63 divided by 7 equals 9. So, the inequality simplifies to:

$x \le 9$

Step 4: Interpret the Solution

The final inequality $x \le 9$ means that $x$ can be any number less than or equal to 9. This is the solution to the inequality. In other words, any value of $x$ that is 9 or smaller will satisfy the original inequality $7x \le 63$.

Step 5: Verify the Solution

To ensure the solution is correct, you can test a few values of $x$ that are less than or equal to 9. For example, if $x = 9$, substituting it back into the original inequality gives:

$7(9) \le 63$

$63 \le 63$

This is true. You can also test a value less than 9, such as $x = 5$:

$7(5) \le 63$

$35 \le 63$

This is also true. Therefore, the solution $x \le 9$ is verified.

Here are some frequently asked questions about solving one-step linear inequalities:

1. What is a one-step linear inequality?

   A one-step linear inequality is a mathematical statement that compares two expressions using inequality symbols (<, >, , ) and can be solved in one step. For example, x + 3 > 7 is a one-step linear inequality.

2. How do you solve a one-step linear inequality?

   To solve a one-step linear inequality, perform the inverse operation on both sides of the inequality to isolate the variable. For example, to solve x + 3 > 7, subtract 3 from both sides: x > 4.

3. What happens when you multiply or divide an inequality by a negative number?

   When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. For example, if -2x < 6, dividing both sides by -2 gives x > -3.

4. How do you graph the solution of a one-step linear inequality on a number line?

   To graph the solution, use an open circle () for strict inequalities (< or >) and a closed circle () for inclusive inequalities ( or ) at the boundary point. Then, shade the line in the direction that satisfies the inequality.

5. Can you give an example of a real-life application of one-step linear inequalities?

   A real-life application could be determining how many hours you need to work to earn at least a certain amount of money. For instance, if you earn $15 per hour and want to make at least $300, you can set up the inequality 15x 300, where x is the number of hours worked.

Understanding the foundation of mathematics is crucial when tackling more advanced concepts like solving one-step linear inequalities. To excel in this area, it's essential to grasp several key prerequisite topics that form the building blocks of algebraic problem-solving.

One fundamental skill is solving equations, particularly those involving distance and time. This ability helps students develop a strong sense of algebraic reasoning and equation manipulation, which directly translates to working with inequalities. By mastering these types of problems, students can more easily transition to understanding the subtle differences between equations and inequalities.

While one-step linear inequalities are relatively straightforward, it's beneficial to have exposure to more complex concepts like graphing inequalities. This broader perspective allows students to visualize how inequalities work on a coordinate plane, reinforcing their understanding of the number line and inequality symbols used in simpler problems.

Additionally, familiarity with compound inequalities can provide valuable context. Although one-step inequalities are less complex, understanding how multiple inequalities can be combined helps students appreciate the versatility and importance of inequality concepts in mathematics.

A solid grasp of basic arithmetic operations is also crucial. Specifically, proficiency in multiplying by negative numbers and dividing by negative numbers is essential when solving inequalities. These operations often come into play when manipulating inequality expressions, and understanding how they affect the inequality sign is vital for arriving at correct solutions.

By mastering these prerequisite topics, students build a strong foundation for tackling one-step linear inequalities. The ability to solve equations provides the basic algebraic skills needed, while exposure to graphing and compound inequalities offers a broader context. Proficiency in working with negative numbers in multiplication and division ensures students can confidently manipulate inequality expressions.

As students progress through these prerequisite topics, they develop critical thinking skills and mathematical intuition that are invaluable when solving one-step linear inequalities. Each concept builds upon the last, creating a comprehensive understanding of algebraic relationships and inequalities. This solid foundation not only aids in solving one-step linear inequalities but also prepares students for more advanced mathematical concepts they will encounter in their future studies.

In conclusion, taking the time to thoroughly understand these prerequisite topics will greatly enhance a student's ability to solve one-step linear inequalities with confidence and accuracy. It's an investment in mathematical knowledge that pays dividends throughout one's academic journey and beyond.

• Evaluating algebraic expressions
• Solving linear equations using multiplication and division
• Solving two-step linear equations: ax + b = c, x/a + b = c
• Solving linear equations using distributive property: a(x + b) = c

• Graphing linear inequalities in two variables
• Graphing systems of linear inequalities
• Graphing quadratic inequalities in two variables
• Graphing systems of quadratic inequalities