Solving Absolute Value Inequalities A Step By Step Guide To $|2x-9| \leq 2$

In the realm of mathematics, especially when dealing with inequalities, absolute values often present a unique challenge. Understanding how to solve absolute value inequalities is crucial for various mathematical applications. This article delves into a step-by-step solution for the inequality $|2x-9| \leq 2$, ultimately expressing the solution in interval notation. We will explore the fundamental principles behind absolute value inequalities and demonstrate how to apply these principles to arrive at the correct answer. The process involves transforming the absolute value inequality into a compound inequality, solving for the variable, and finally, representing the solution set in interval form. Understanding these steps is essential for anyone studying algebra or related fields.

The absolute value of a number represents its distance from zero on the number line. Therefore, when we encounter an inequality involving an absolute value, we are essentially dealing with a range of values rather than a single point. For instance, $|x| < a$ implies that the distance of $x$ from zero is less than $a$, which translates to $-a < x < a$. Similarly, $|x| > a$ means that the distance of $x$ from zero is greater than $a$, leading to two separate inequalities: $x < -a$ or $x > a$. These basic principles are the building blocks for solving more complex absolute value inequalities. The key is to recognize that an absolute value inequality can be converted into a compound inequality, which can then be solved using standard algebraic techniques. By understanding the underlying concepts, we can confidently tackle a wide variety of absolute value problems.

When dealing with absolute value inequalities, it's also important to remember the properties of inequalities themselves. For example, multiplying or dividing both sides of an inequality by a negative number requires flipping the inequality sign. This rule is crucial for maintaining the correctness of the solution. Additionally, understanding how to manipulate inequalities, such as adding or subtracting the same value from both sides, is essential for isolating the variable. The combination of absolute value principles and inequality properties forms the foundation for solving these types of problems. With practice and a solid grasp of the concepts, solving absolute value inequalities becomes a straightforward and manageable task.

The given inequality is $|2x-9| \leq 2$. To solve this, we must first understand the meaning of the absolute value. The absolute value of a quantity is its distance from zero, so $|2x-9| \leq 2$ means that the distance between $2x-9$ and zero is less than or equal to 2. This translates to a compound inequality, where $2x-9$ is bounded between -2 and 2. This transformation is a critical step in solving absolute value inequalities, as it allows us to remove the absolute value signs and work with more familiar algebraic expressions. The compound inequality effectively captures the two possible scenarios: $2x-9$ can be a positive value close to zero, or it can be a negative value close to zero. Both scenarios are encompassed within the inequality $|2x-9| \leq 2$.

To proceed, we rewrite the absolute value inequality as a compound inequality: $-2 \leq 2x-9 \leq 2$. This compound inequality states that $2x-9$ must be greater than or equal to -2 and less than or equal to 2. This is the crucial step in converting the absolute value problem into a more manageable form. The ability to make this transformation is fundamental to solving any absolute value inequality. Once the absolute value is removed, we can apply standard algebraic techniques to isolate the variable $x$. This transformation simplifies the problem and allows us to find the range of values for $x$ that satisfy the original inequality.

The compound inequality $-2 \leq 2x-9 \leq 2$ effectively represents the range of possible values for $2x-9$. It is essential to remember that this transformation is valid because of the definition of absolute value. By understanding this connection, we can confidently apply this technique to solve similar problems. The next step involves isolating $x$ in the compound inequality, which will give us the solution set. This process involves performing the same operations on all three parts of the inequality, ensuring that the inequality remains balanced. By carefully following these steps, we can accurately determine the solution to the absolute value inequality.

Now, let's solve the compound inequality $-2 \leq 2x-9 \leq 2$. Our goal is to isolate $x$ in the middle. To do this, we first add 9 to all three parts of the inequality: $-2 + 9 \leq 2x - 9 + 9 \leq 2 + 9$. This simplifies to $7 \leq 2x \leq 11$. This step is crucial in isolating the term with $x$ in the middle of the compound inequality. By adding 9 to all parts, we maintain the balance of the inequality and move closer to isolating $x$. This is a standard algebraic technique used to solve inequalities, and it's essential for solving compound inequalities as well.

Next, we divide all three parts of the inequality by 2: $\frac{7}{2} \leq \frac{2x}{2} \leq \frac{11}{2}$. This simplifies to $\frac{7}{2} \leq x \leq \frac{11}{2}$. This step completes the isolation of $x$ and provides the solution to the inequality. By dividing all parts of the inequality by 2, we determine the range of values for $x$ that satisfy the original inequality. This process is a straightforward application of algebraic principles and demonstrates the importance of maintaining balance when manipulating inequalities. The solution $\frac{7}{2} \leq x \leq \frac{11}{2}$ represents a continuous range of values for $x$, which can be expressed in interval notation.

The solution $\frac{7}{2} \leq x \leq \frac{11}{2}$ tells us that $x$ can be any value between $\frac{7}{2}$ and $\frac{11}{2}$, inclusive. This means that $x$ can be equal to $\frac{7}{2}$, equal to $\frac{11}{2}$, or any number in between. Understanding this range is crucial for expressing the solution in interval notation. The inequality provides a clear and concise representation of the possible values for $x$, and it is the foundation for the final step of expressing the solution as an interval. By accurately solving the compound inequality, we have determined the range of values that satisfy the original absolute value inequality.

The solution $\frac{7}{2} \leq x \leq \frac{11}{2}$ indicates that $x$ is greater than or equal to $\frac{7}{2}$ and less than or equal to $\frac{11}{2}$. In interval notation, this is represented as [rac{7}{2}, rac{11}{2}]. The square brackets indicate that the endpoints $\frac{7}{2}$ and $\frac{11}{2}$ are included in the solution set. Interval notation is a concise way to express a range of values, and it is commonly used in mathematics to represent solutions to inequalities. The use of brackets and parentheses is crucial in interval notation, as they indicate whether the endpoints are included or excluded from the solution set. In this case, the square brackets signify that the endpoints are part of the solution.

The interval [rac{7}{2}, rac{11}{2}] represents all real numbers between $\frac{7}{2}$ and $\frac{11}{2}$, including the endpoints. This interval is a closed interval, as it includes both endpoints. In contrast, an open interval would be represented using parentheses, such as $(a, b)$, which indicates all real numbers between $a$ and $b$, excluding $a$ and $b$. Understanding the different types of intervals and their notation is essential for correctly expressing solutions to inequalities. The interval notation provides a clear and unambiguous representation of the solution set.

Therefore, the final answer to the inequality $|2x-9| \leq 2$ is the interval [rac{7}{2}, rac{11}{2}]. This interval represents all values of $x$ that satisfy the original inequality. The process of solving the absolute value inequality involved transforming it into a compound inequality, solving for $x$, and then expressing the solution in interval notation. This step-by-step approach demonstrates the importance of understanding the underlying principles of absolute value and inequalities. By accurately applying these principles, we can confidently solve a wide range of mathematical problems.

Final Answer: The final answer is $\boxed{[7/2, 11/2]}$