Solving Inequalities A Comprehensive Guide To -5x + 7 > 42

Jul 9, 2025 by ADMIN 59 views

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. This article delves into the intricacies of solving the linear inequality $-5x + 7 > 42$ , providing a step-by-step approach and exploring the underlying principles. Mastering the techniques for solving inequalities is essential for various mathematical applications, including optimization problems, calculus, and real-world scenarios. Let's embark on this journey to unravel the solution to this inequality and gain a deeper understanding of the concepts involved.

Understanding Linear Inequalities

Before diving into the specific solution, it's vital to grasp the fundamental concepts of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as >, <, ≥, or ≤. Unlike equations, which assert the equality of two expressions, inequalities indicate a range of possible values that satisfy the given condition. Solving an inequality involves isolating the variable on one side of the inequality symbol to determine the set of values that make the inequality true.

The inequality $-5x + 7 > 42$ is a linear inequality because the variable x is raised to the power of 1. To solve it, we need to manipulate the inequality while preserving its truth value. This involves applying algebraic operations such as addition, subtraction, multiplication, and division, with a crucial rule to remember: multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

Step-by-Step Solution of $-5x + 7 > 42$

Now, let's systematically solve the inequality $-5x + 7 > 42$ . We will follow a step-by-step approach, explaining each operation and its effect on the inequality.

Step 1: Isolate the term with x

The first step is to isolate the term containing the variable x. In this case, we need to isolate $-5x$ . To do this, we subtract 7 from both sides of the inequality. This operation maintains the balance of the inequality, just like subtracting the same weight from both sides of a balance scale.

$-5x + 7 - 7 > 42 - 7$

This simplifies to:

$-5x > 35$

Step 2: Isolate x

Next, we need to isolate x completely. Since x is multiplied by -5, we divide both sides of the inequality by -5. Here's where the critical rule comes into play: dividing by a negative number reverses the inequality symbol. The "greater than" (>) symbol will become a "less than" (<) symbol.

$rac{-5x}{-5} < rac{35}{-5}$

This simplifies to:

$x < -7$

Step 3: Interpret the solution

The solution to the inequality is $x < -7$ . This means that any value of x that is less than -7 will satisfy the original inequality $-5x + 7 > 42$ . For instance, if we substitute x = -8 into the original inequality, we get:

$-5(-8) + 7 > 42$

$40 + 7 > 42$

$47 > 42$

This is a true statement, confirming that -8 is indeed a solution. Conversely, if we substitute x = -6, which is greater than -7, we get:

$-5(-6) + 7 > 42$

$30 + 7 > 42$

$37 > 42$

This is a false statement, demonstrating that -6 is not a solution.

Representing the Solution on a Number Line

Visualizing the solution on a number line can provide a clearer understanding of the range of values that satisfy the inequality. To represent $x < -7$ on a number line, we draw a number line and mark -7. Since the inequality is strictly less than (-7), we use an open circle at -7 to indicate that -7 itself is not included in the solution. Then, we shade the region to the left of -7, representing all values less than -7.

Common Mistakes to Avoid

When solving inequalities, it's crucial to avoid common pitfalls that can lead to incorrect solutions. Here are some frequent mistakes to watch out for:

Forgetting to reverse the inequality symbol: The most common mistake is failing to reverse the inequality symbol when multiplying or dividing both sides by a negative number. This can lead to an entirely incorrect solution set.
Incorrectly applying the distributive property: If the inequality involves parentheses, ensure you distribute correctly. For example, in the inequality $2(x + 3) > 8$ , you must distribute the 2 to both x and 3.
Making arithmetic errors: Simple arithmetic mistakes can derail the entire solution process. Double-check your calculations at each step to minimize errors.
Misinterpreting the solution: Understand what the solution set represents. For instance, $x < -7$ means all values less than -7, not greater than -7.

Importance of Inequalities in Mathematics and Real-World Applications

Inequalities are not just abstract mathematical concepts; they have significant applications in various fields, including:

Optimization problems: Inequalities are used to define constraints in optimization problems, helping to find the maximum or minimum values of a function subject to certain conditions.
Calculus: Inequalities play a crucial role in calculus, particularly in the study of limits, continuity, and the behavior of functions.
Economics: Inequalities are used to model economic constraints, such as budget limitations and resource scarcity.
Engineering: Inequalities are used in engineering design to ensure that structures and systems meet safety and performance requirements.
Computer science: Inequalities are used in algorithms and data structures, such as sorting algorithms and search algorithms.

Conclusion

Solving inequalities is a fundamental skill in mathematics with far-reaching applications. By understanding the principles and techniques involved, you can confidently tackle various inequality problems. Remember the crucial rule of reversing the inequality symbol when multiplying or dividing by a negative number, and always double-check your work to avoid common mistakes. The solution to the inequality $-5x + 7 > 42$ is $x < -7$ , representing all values less than -7. Mastering inequalities will undoubtedly enhance your mathematical prowess and open doors to a deeper understanding of the world around you.

Choosing the Correct Answer

Based on our step-by-step solution, we found that $x < -7$ . Now, let's compare this result with the given options:

A. $x > 245$ B. $x < -175$ C. $x > 7$ D. $x < -7$

Our solution matches option D, $x < -7$ . Therefore, the correct answer is D.

By carefully following the steps and understanding the underlying principles, we have successfully solved the inequality and identified the correct solution.