Solving 2x - 6 ≥ 6(x - 2) + 8 Finding The Number Line Solution

In the realm of mathematics, inequalities play a crucial role in defining relationships between expressions. Specifically, inequalities allow us to express scenarios where one value is greater than, less than, or not equal to another. In this article, we will delve into the process of solving a given inequality and representing its solution set on a number line. Our focus will be on the inequality $2x - 6 ≥ 6(x - 2) + 8$, which presents an engaging exercise in algebraic manipulation and the interpretation of solutions.

Understanding Inequalities

Before we embark on the solution process, it is essential to grasp the fundamental concept of inequalities. Unlike equations that seek specific values that satisfy a relationship, inequalities define a range of values that fulfill a particular condition. The symbols used in inequalities, such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤), each carry distinct meanings. Understanding these symbols is paramount to accurately interpreting the solution set of an inequality.

Inequalities often arise in real-world scenarios, such as determining the minimum score needed to achieve a certain grade, calculating the maximum weight a bridge can support, or establishing the budget constraints for a project. Their versatility makes them a cornerstone of mathematical problem-solving.

The Inequality $2x - 6 ≥ 6(x - 2) + 8$

The specific inequality we aim to solve, $2x - 6 ≥ 6(x - 2) + 8$, involves a linear expression on both sides of the inequality symbol. This type of inequality is commonly encountered in algebra and can be solved using a series of algebraic manipulations, mirroring the steps involved in solving linear equations. However, a crucial distinction lies in how we handle operations that involve multiplying or dividing by a negative number, as this can affect the direction of the inequality sign.

Solving the Inequality Step-by-Step

To unravel the solution set for the inequality $2x - 6 ≥ 6(x - 2) + 8$, we will embark on a step-by-step journey, employing algebraic techniques to isolate the variable x. This process will involve simplifying expressions, combining like terms, and carefully managing the inequality sign.

Step 1: Distribute the constants

The first step in our algebraic endeavor is to distribute the constants on both sides of the inequality. This involves multiplying the constant outside the parentheses by each term inside the parentheses. In our case, we have:

2x - 6 ≥ 6(x - 2) + 8$ simplifies to $2x - 6 ≥ 6x - 12 + 8$. ### Step 2: Combine Like Terms Next, we aim to simplify the expressions on each side of the inequality by combining like terms. This involves identifying terms with the same variable or constant component and adding or subtracting their coefficients. On the right side of our inequality, we have two constant terms, -12 and +8, which can be combined: $2x - 6 ≥ 6x - 12 + 8$ becomes $2x - 6 ≥ 6x - 4$. ### Step 3: Isolate the variable term Our objective is to isolate the variable term (the term containing x) on one side of the inequality. To achieve this, we can subtract 2x from both sides of the inequality. This ensures that we maintain the balance of the inequality while moving the variable term to one side: $2x - 6 ≥ 6x - 4$ becomes $-6 ≥ 4x - 4$. ### Step 4: Isolate the variable To further isolate the variable x, we need to eliminate the constant term on the same side as the variable. We can accomplish this by adding 4 to both sides of the inequality: $-6 ≥ 4x - 4$ becomes $-2 ≥ 4x$. ### Step 5: Solve for the variable The final step in solving for x is to divide both sides of the inequality by the coefficient of x, which is 4 in our case. However, it is crucial to remember that when we divide both sides of an inequality by a negative number, we must flip the direction of the inequality sign. Since we are dividing by a positive number (4), the inequality sign remains the same: $-2 ≥ 4x$ becomes $x ≤ -\frac{1}{2}$. ## Interpreting the Solution Set The solution to our inequality, $x ≤ -\frac{1}{2}$, tells us that any value of x that is less than or equal to -1/2 will satisfy the original inequality. This solution set can be represented in various ways, including set notation, interval notation, and graphically on a number line. ### Set Notation In set notation, the solution set is expressed as: $\{x \mid x ≤ -\frac{1}{2}\}$. This notation reads as "the set of all x such that x is less than or equal to -1/2." ### Interval Notation Interval notation provides a concise way to represent the solution set using intervals. For our inequality, the interval notation is: $(-\infty, -\frac{1}{2}]$. The parenthesis on the left indicates that negative infinity is not included in the solution set (as it is a concept rather than a specific number), while the square bracket on the right indicates that -1/2 is included in the solution set. ### Graphical Representation on a Number Line The most visually intuitive way to represent the solution set is by using a number line. A number line is a straight line on which numbers are marked at intervals. To represent our solution set, we follow these steps: 1. Draw a number line and mark the critical value, -1/2, on the line. 2. Since our solution includes values less than or equal to -1/2, we draw a closed circle (or a solid dot) at -1/2 to indicate that this value is part of the solution set. 3. Draw an arrow extending from the closed circle to the left, indicating that all values to the left of -1/2 are also part of the solution set. ## Identifying the Correct Number Line Representation When presented with multiple number lines, each depicting a different range of values, we can identify the correct representation of our solution set by comparing the critical value and the direction of the arrow. The correct number line should have a closed circle at -1/2 and an arrow pointing to the left, signifying that all values less than or equal to -1/2 are solutions to the inequality. ## Common Mistakes to Avoid While solving inequalities, it is crucial to be mindful of potential pitfalls that can lead to incorrect solutions. Some common mistakes include: * **Forgetting to flip the inequality sign:** As mentioned earlier, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. Failing to do so will result in an incorrect solution set. * **Incorrectly distributing constants:** Ensure that the constant outside the parentheses is multiplied by every term inside the parentheses. A mistake in distribution can alter the entire equation and lead to a wrong answer. * **Combining unlike terms:** Only like terms (terms with the same variable or constant component) can be combined. Combining unlike terms will lead to an incorrect simplification of the expression. * **Misinterpreting the number line representation:** Pay close attention to the type of circle (open or closed) at the critical value and the direction of the arrow. An open circle indicates that the critical value is not included in the solution set, while a closed circle indicates that it is included. ## Practice Problems To solidify your understanding of solving inequalities and representing their solutions on number lines, try solving the following practice problems: 1. Solve the inequality $3x + 5 < 14$ and represent the solution on a number line. 2. Solve the inequality $-2(x - 1) ≥ 8$ and represent the solution on a number line. 3. Solve the inequality $4x - 7 ≤ 2x + 3$ and represent the solution on a number line. By working through these problems, you will reinforce your skills and gain confidence in your ability to tackle inequalities. ## Conclusion Solving inequalities is a fundamental skill in algebra, with applications spanning various fields. By mastering the steps involved in solving inequalities and understanding how to represent their solutions on number lines, you will be well-equipped to tackle a wide range of mathematical problems. Remember to pay close attention to the inequality sign, distribute constants correctly, and avoid common mistakes. With practice and perseverance, you can confidently navigate the world of inequalities. # Keywords **Inequalities**, **solution set**, **number line**, **algebraic manipulation**, **linear expression**, **critical value**, **set notation**, **interval notation**, **graphical representation**, **common mistakes**, **practice problems**.