Solving The Inequality 1/8 X ≤ 1/2 X + 15 A Step-by-Step Guide

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In the realm of mathematics, inequalities play a crucial role in defining relationships between quantities that are not necessarily equal. Solving inequalities is a fundamental skill, allowing us to determine the range of values that satisfy a given condition. This article provides a comprehensive guide to solving the inequality $\frac{1}{8}x \leq \frac{1}{2}x + 15$, offering step-by-step instructions and explanations to ensure a clear understanding of the process.

Understanding Inequalities

Before diving into the solution, it's essential to grasp the concept of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another. The symbols used to represent these relationships are:

• < (less than)

• (greater than)

• $\leq$ (less than or equal to)
• $\geq$ (greater than or equal to)

Inequalities are used extensively in various mathematical and real-world applications, from optimizing resources to modeling constraints. Understanding how to solve them is therefore an indispensable skill.

Step-by-Step Solution of $\frac{1}{8}x \leq \frac{1}{2}x + 15$

To solve the inequality $\frac{1}{8}x \leq \frac{1}{2}x + 15$, we will follow a series of algebraic manipulations to isolate the variable $x$ on one side of the inequality. This process is similar to solving equations, with a few key differences to be mindful of. Let's break down the solution step by step:

1. Eliminate Fractions

The presence of fractions can make the inequality appear more complex than it is. The first step is to eliminate these fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators. In this case, the denominators are 8 and 2, and their LCM is 8. Multiplying both sides by 8, we get:

$8 \times \frac{1}{8}x \leq 8 \times (\frac{1}{2}x + 15)$

This simplifies to:

$x \leq 4x + 120$

2. Group the Variable Terms

Next, we want to gather all the terms containing the variable $x$ on one side of the inequality. To do this, we subtract $4x$ from both sides:

$x - 4x \leq 4x + 120 - 4x$

This simplifies to:

$-3x \leq 120$

3. Isolate the Variable

Now, we need to isolate $x$ by dividing both sides of the inequality by the coefficient of $x$, which is -3. However, there's a crucial rule to remember when dealing with inequalities: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the quantities involved, which can alter the relationship between them.

So, dividing both sides by -3 and reversing the inequality sign, we get:

$\frac{-3x}{-3} \geq \frac{120}{-3}$

This simplifies to:

$x \geq -40$

4. Interpret the Solution

The solution $x \geq -40$ means that any value of $x$ that is greater than or equal to -40 will satisfy the original inequality. This represents an interval of values on the number line, starting from -40 and extending to positive infinity.

Representing the Solution

There are several ways to represent the solution to an inequality:

1. Interval Notation

Interval notation is a concise way to express a range of values. For the solution $x \geq -40$, the interval notation is $[-40, \infty)$. The square bracket on the left indicates that -40 is included in the solution, while the parenthesis on the right indicates that infinity is not a specific number and is therefore not included.

2. Graph on a Number Line

Another way to visualize the solution is to graph it on a number line. We draw a closed circle (or a square bracket) at -40 to indicate that it is included in the solution, and then shade the line to the right, representing all values greater than -40.

3. Set-Builder Notation

Set-builder notation is a more formal way to express the solution as a set. For $x \geq -40$, the set-builder notation is:

$\lbrace x \mid x \geq -40 \rbrace$

This is read as "the set of all $x$ such that $x$ is greater than or equal to -40."

Common Mistakes to Avoid

When solving inequalities, it's important to be aware of common pitfalls that can lead to incorrect solutions. Here are a few mistakes to watch out for:

1. Forgetting to Reverse the Inequality Sign

As mentioned earlier, when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is perhaps the most common mistake in solving inequalities. Forgetting to do this will result in an incorrect solution.

2. Incorrectly Distributing

When dealing with inequalities that involve parentheses, it's crucial to distribute correctly. Make sure to multiply the term outside the parentheses by each term inside the parentheses. A mistake in distribution can lead to an entirely different inequality and an incorrect solution.

3. Combining Unlike Terms

Just like in equations, you can only combine like terms in inequalities. This means you can add or subtract terms with the same variable and exponent, but you cannot combine terms with different variables or exponents. Mixing up unlike terms will lead to an incorrect simplification and an incorrect solution.

4. Misinterpreting the Solution

Once you've solved the inequality, it's important to correctly interpret the solution. Understand whether the solution includes the endpoint (using square brackets or closed circles) or excludes it (using parentheses or open circles). Misinterpreting the solution can lead to applying the results incorrectly.

Applications of Inequalities

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

1. Optimization Problems

Inequalities are used extensively in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints. These constraints are often expressed as inequalities. For instance, a business might want to maximize its profit while staying within its budget, which can be modeled using inequalities.

2. Resource Allocation

Inequalities are also used in resource allocation problems, where the goal is to distribute resources in the most efficient way. For example, a factory might have a limited amount of raw materials and needs to decide how much of each product to produce to maximize its output, subject to the constraint of available resources.

3. Constraints and Boundaries

In many situations, there are natural constraints or boundaries that can be expressed as inequalities. For instance, the speed of a car must be within certain limits, or the temperature of a room must be within a comfortable range. Inequalities can be used to define these limits and ensure that conditions are met.

4. Modeling Uncertainty

Inequalities can also be used to model uncertainty. For example, if we know that a certain quantity is within a certain range but we don't know its exact value, we can express this using an inequality. This is particularly useful in statistics and probability.

Practice Problems

To solidify your understanding of solving inequalities, here are a few practice problems:

1. Solve for $x$: $3x - 5 > 7$
2. Solve for $y$: $-2y + 4 \leq 10$
3. Solve for $z$: $\frac{1}{3}z + 2 < 5$
4. Solve for $w$: $4w - 1 \geq 2w + 3$

By working through these problems, you can reinforce the concepts and techniques discussed in this article and develop your problem-solving skills.

Conclusion

Solving inequalities is a crucial skill in mathematics with wide-ranging applications. This article has provided a step-by-step guide to solving the inequality $\frac{1}{8}x \leq \frac{1}{2}x + 15$, along with explanations of the underlying concepts and common mistakes to avoid. By understanding the principles of inequalities and practicing regularly, you can confidently tackle a variety of problems and apply these skills in real-world scenarios. Remember the key rules, such as reversing the inequality sign when multiplying or dividing by a negative number, and you'll be well on your way to mastering inequalities. Keep practicing, and you'll find that solving inequalities becomes second nature! We hope this comprehensive guide has been helpful in your mathematical journey. Whether you are a student learning the basics or someone looking to refresh your skills, understanding inequalities is a valuable asset.