Solving The Inequality -2x - 8 > 3x + 12 A Step-by-Step Guide

$$

This article delves into solving the linear inequality $-2x - 8 > 3x + 12$, providing a step-by-step solution and explaining the underlying concepts. Understanding inequalities is crucial in mathematics, as they form the basis for various applications in fields like optimization, calculus, and economics. This guide will not only solve the given inequality but also offer a broader understanding of linear inequalities and their solutions.

Introduction to Linear Inequalities

Linear inequalities are mathematical expressions that compare two values using inequality symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike linear equations, which have a single solution, linear inequalities typically have a range of solutions. Solving linear inequalities involves isolating the variable on one side of the inequality, similar to solving linear equations, but with a crucial difference: multiplying or dividing by a negative number reverses the inequality sign. This article aims to provide a detailed explanation of how to solve the given inequality, ensuring a clear understanding of the process and the underlying principles.

Understanding inequalities is fundamental in various areas of mathematics and its applications. For instance, in calculus, inequalities are used to define intervals and analyze the behavior of functions. In optimization problems, inequalities are used to set constraints and find optimal solutions. Even in everyday life, inequalities are used to make comparisons and decisions, such as determining if a budget is sufficient or if a certain condition is met. Therefore, mastering the techniques for solving inequalities is not just an academic exercise but a practical skill with real-world relevance.

Step-by-Step Solution of $-2x - 8 > 3x + 12$

To solve the inequality $-2x - 8 > 3x + 12$, we will follow a series of algebraic steps to isolate the variable x on one side of the inequality. This process is similar to solving linear equations, but we must remember the rule about reversing the inequality sign when multiplying or dividing by a negative number. Here’s a detailed breakdown of each step:

Step 1: Combine Like Terms

The first step is to group the terms with x on one side of the inequality and the constant terms on the other side. We can do this by adding $2x$ to both sides of the inequality to eliminate the $-2x$ term on the left side. This gives us:

$-2x - 8 + 2x > 3x + 12 + 2x$

Simplifying this, we get:

$-8 > 5x + 12$

Step 2: Isolate the Variable Term

Next, we want to isolate the term with x (which is $5x$) on one side of the inequality. To do this, we subtract 12 from both sides:

$-8 - 12 > 5x + 12 - 12$

This simplifies to:

$-20 > 5x$

Step 3: Solve for x

Now, we need to solve for x by dividing both sides of the inequality by 5. Since 5 is a positive number, we do not need to reverse the inequality sign:

$\frac{-20}{5} > \frac{5x}{5}$

This simplifies to:

$-4 > x$

Step 4: Rewrite the Inequality

It is common practice to write the inequality with the variable on the left side. To do this, we can rewrite $-4 > x$ as $x < -4$. This means that x is less than -4.

Therefore, the solution to the inequality $-2x - 8 > 3x + 12$ is $x < -4$.

Understanding the Solution Set

The solution set for the inequality $x < -4$ includes all real numbers less than -4. This can be represented graphically on a number line by shading the region to the left of -4. The open circle at -4 indicates that -4 is not included in the solution set. In interval notation, the solution set is written as $(-\infty, -4)$.

It’s important to understand that an inequality typically has a range of solutions, unlike an equation which usually has a single solution. In this case, any value of x that is less than -4 will satisfy the original inequality. For example, if we substitute $x = -5$ into the original inequality, we get:

$-2(-5) - 8 > 3(-5) + 12$

$10 - 8 > -15 + 12$

$2 > -3$

This is a true statement, confirming that $x = -5$ is indeed a solution. Conversely, if we substitute $x = -3$ (which is not less than -4), we get:

$-2(-3) - 8 > 3(-3) + 12$

$6 - 8 > -9 + 12$

$-2 > 3$

This is a false statement, showing that $x = -3$ is not a solution.

Common Mistakes to Avoid

When solving inequalities, it is easy to make mistakes, especially when dealing with negative numbers. Here are some common mistakes to avoid:

1. Forgetting to Reverse the Inequality Sign: The most common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides of the inequality by a negative number. Remember, this step is crucial for obtaining the correct solution.
2. Incorrectly Combining Like Terms: Make sure to combine like terms correctly. Double-check your arithmetic when adding or subtracting terms on both sides of the inequality.
3. Misinterpreting the Solution Set: Understand what the solution set represents. For instance, $x < -4$ means all numbers less than -4, not greater than -4. Representing the solution on a number line can help visualize the correct range.
4. Not Checking the Solution: Always check your solution by substituting a value from the solution set back into the original inequality. This can help identify any errors in your calculations.

Real-World Applications of Linear Inequalities

Linear inequalities are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

1. Budgeting

When budgeting, you often need to ensure that your expenses do not exceed your income. This can be represented as an inequality. For example, if your income is $2000 per month and your expenses are represented by the expression $500 + 200x$, where x is the number of discretionary items you purchase, the inequality $500 + 200x ≤ 2000$ ensures that your expenses stay within your income.

2. Speed Limits

Speed limits on roads are a practical application of inequalities. A speed limit sign that reads “65 mph” means that your speed, s, must satisfy the inequality $s ≤ 65$.

3. Manufacturing

In manufacturing, inequalities are used to set tolerance limits. For example, if a machine produces parts that are supposed to be 10 cm long, a tolerance of ±0.1 cm might be allowed. This means that the actual length, L, of the part must satisfy the inequality $9.9 ≤ L ≤ 10.1$.

4. Nutrition

Nutrition guidelines often use inequalities. For example, a guideline might state that you should consume no more than 2000 calories per day. If c represents your daily calorie intake, this guideline can be expressed as $c ≤ 2000$.

5. Optimization Problems

In optimization problems, such as linear programming, inequalities are used to define constraints. These constraints limit the possible solutions, and the goal is to find the optimal solution within these constraints. For example, a company might want to maximize its profit subject to constraints on resources and production capacity.

Conclusion

In conclusion, solving the inequality $-2x - 8 > 3x + 12$ involves a series of algebraic steps, resulting in the solution $x < -4$. This solution represents all real numbers less than -4. Understanding the process of solving linear inequalities is essential not only for academic purposes but also for practical applications in various fields. By mastering the techniques discussed in this article and avoiding common mistakes, you can confidently solve linear inequalities and apply them to real-world problems. Remember the key steps: combine like terms, isolate the variable term, solve for the variable, and, most importantly, reverse the inequality sign when multiplying or dividing by a negative number. With practice and a solid understanding of the underlying principles, solving linear inequalities will become a straightforward task.

This article has provided a comprehensive guide to solving linear inequalities, covering the step-by-step solution of the example inequality, common mistakes to avoid, and real-world applications. By understanding these concepts, you can enhance your mathematical skills and apply them effectively in various contexts.