Solving Inequalities Finding The Value Of X In 4x - 12 ≤ 16 + 8x

In this article, we will delve into the process of solving the inequality $4x - 12 \leq 16 + 8x$. This is a fundamental concept in algebra, and mastering it is crucial for success in higher-level mathematics. We will break down the steps involved in solving this inequality, explain the underlying principles, and discuss how to interpret the solution set. By the end of this article, you will have a clear understanding of how to solve similar inequalities and confidently identify the correct solution.

Before we dive into the specific problem, let's first understand what an inequality is. An inequality is a mathematical statement that compares two expressions using symbols such as less than (<), greater than (>), less than or equal to ($\leq$), or greater than or equal to ($\geq$). Unlike an equation, which asserts that two expressions are equal, an inequality indicates a range of possible values that satisfy the given condition. Solving an inequality involves finding all values of the variable that make the inequality true. These values form the solution set, which can be represented graphically on a number line or expressed in interval notation. When dealing with inequalities, it's important to remember that certain operations, such as multiplying or dividing by a negative number, require flipping the inequality sign to maintain the truth of the statement.

To solve the inequality $4x - 12 \leq 16 + 8x$, we need to isolate the variable $x$ on one side of the inequality. This involves performing algebraic operations on both sides while maintaining the inequality's balance. Let's break down the solution into a series of steps:

1. Combine Like Terms

The first step is to gather all the terms containing $x$ on one side of the inequality and the constant terms on the other side. To do this, we can subtract $4x$ from both sides of the inequality:

$4x - 12 - 4x \leq 16 + 8x - 4x$

This simplifies to:

$-12 \leq 16 + 4x$

Next, we subtract $16$ from both sides to isolate the term with $x$:

$-12 - 16 \leq 16 + 4x - 16$

This simplifies to:

$-28 \leq 4x$

2. Isolate the Variable

Now, we need to isolate $x$ by dividing both sides of the inequality by the coefficient of $x$, which is $4$. Since we are dividing by a positive number, we do not need to flip the inequality sign:

$\frac{-28}{4} \leq \frac{4x}{4}$

This simplifies to:

$-7 \leq x$

This inequality can also be written as:

$x \geq -7$

3. Interpret the Solution

The solution $x \geq -7$ means that any value of $x$ that is greater than or equal to $-7$ will satisfy the original inequality. This includes $-7$ itself, as well as any number to the right of $-7$ on the number line.

Now that we have the solution set, let's examine the answer choices provided:

A. -10 B. -9 C. -8 D. -7

We are looking for a value of $x$ that is in the solution set, meaning it must be greater than or equal to $-7$. Let's analyze each option:

• A. -10: This value is less than $-7$, so it is not in the solution set.
• B. -9: This value is also less than $-7$, so it is not in the solution set.
• C. -8: This value is less than $-7$, so it is not in the solution set.
• D. -7: This value is equal to $-7$, so it is in the solution set because our solution includes values greater than or equal to -7.

Therefore, the correct answer is D. -7.

To further illustrate the solution, we can represent the inequality $x \geq -7$ on a number line. We draw a closed circle (or a bracket) at $-7$ to indicate that $-7$ is included in the solution set. Then, we shade the region to the right of $-7$ to represent all values greater than $-7$. This visual representation helps to solidify the concept of a solution set and provides a clear picture of the range of values that satisfy the inequality.

When solving inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and arrive at the correct solution:

1. Forgetting to Flip the Inequality Sign

The most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Remember that when you multiply or divide by a negative number, you must reverse the direction of the inequality to maintain the truth of the statement. For example, if you have $-2x < 4$, dividing both sides by $-2$ gives $x > -2$, not $x < -2$.

2. Incorrectly Combining Like Terms

Another common mistake is incorrectly combining like terms. Make sure to only combine terms that have the same variable and exponent. For example, you can combine $3x$ and $5x$ to get $8x$, but you cannot combine $3x$ and $5x^2$.

3. Misinterpreting the Solution Set

It's also important to correctly interpret the solution set. For example, $x > 3$ means all values greater than 3, but not including 3 itself. $x \geq 3$ means all values greater than or equal to 3, including 3. Pay close attention to the inequality symbol to ensure you understand which values are included in the solution set.

4. Not Checking the Solution

Finally, it's always a good practice to check your solution by plugging a value from your solution set back into the original inequality. If the inequality holds true, then your solution is likely correct. If the inequality does not hold true, then you have made an error and need to re-evaluate your steps.

Inequalities are not just abstract mathematical concepts; they have many real-world applications. They are used in various fields, including:

1. Economics

In economics, inequalities are used to model constraints such as budget limitations. For example, a consumer's spending must be less than or equal to their income, which can be represented as an inequality.

2. Engineering

In engineering, inequalities are used to design structures that can withstand certain loads or to ensure that systems operate within specified limits. For example, the maximum stress on a bridge must be less than a certain value to prevent failure.

3. Computer Science

In computer science, inequalities are used in optimization problems, such as finding the shortest path in a network or minimizing the cost of a project. They are also used in algorithm analysis to determine the efficiency of algorithms.

4. Everyday Life

Inequalities are also used in everyday life. For example, when planning a trip, you might need to ensure that the cost of transportation and accommodation is less than or equal to your budget. Or, when cooking, you might need to ensure that the oven temperature is within a certain range.

Solving inequalities is a fundamental skill in algebra with wide-ranging applications. By understanding the principles behind inequalities and following a systematic approach, you can confidently solve a variety of problems. In this article, we have walked through the steps of solving the inequality $4x - 12 \leq 16 + 8x$, discussed common mistakes to avoid, and explored real-world applications of inequalities. Remember to practice regularly and apply these concepts to different problems to further strengthen your understanding.

In summary, the solution to the inequality $4x - 12 \leq 16 + 8x$ is $x \geq -7$, and the value of $x$ in the solution set from the given options is D. -7. Mastering these skills will not only help you in your math courses but also equip you with valuable problem-solving abilities that can be applied in various aspects of life.