Solving Absolute Value Inequalities A Step-by-Step Guide To 6|x|≥66

In this article, we will delve into solving the inequality $6|x| \geq 66$. This is a mathematical problem that involves absolute values and inequalities. Understanding how to solve such problems is crucial for students and anyone interested in mathematics. This article aims to provide a comprehensive, step-by-step guide to solving this inequality, ensuring clarity and understanding for all readers. We will break down the problem, discuss the concepts involved, and provide a detailed solution. Let's embark on this mathematical journey to unravel the solution to this inequality.

Before diving into the specifics of the inequality $6|x| \geq 66$, it's essential to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line. It is always non-negative. For example, the absolute value of 5 is 5, denoted as $|5| = 5$, and the absolute value of -5 is also 5, denoted as $|-5| = 5$. This is because both 5 and -5 are 5 units away from zero. Mathematically, the absolute value of a number $x$ is defined as:

$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$

This piecewise definition tells us that if $x$ is non-negative, then $|x|$ is simply $x$. If $x$ is negative, then $|x|$ is the negation of $x$, which makes it positive. This concept is crucial because when solving inequalities involving absolute values, we need to consider both cases: when the expression inside the absolute value is positive or zero, and when it is negative. Understanding this distinction is key to correctly solving inequalities like the one we are addressing in this article. The absolute value function is a fundamental concept in mathematics, and it appears in various contexts, from basic algebra to advanced calculus. Its non-negative nature and its property of measuring distance make it a powerful tool in mathematical problem-solving.

Now, let's tackle the inequality $6|x| \geq 66$. Our goal is to find all real numbers $x$ that satisfy this inequality. To do this, we will follow a step-by-step approach. First, we need to isolate the absolute value term. We can do this by dividing both sides of the inequality by 6:

$\frac{6|x|}{6} \geq \frac{66}{6}$

This simplifies to:

$|x| \geq 11$

Now that we have isolated the absolute value, we need to consider the two cases that arise from the definition of absolute value. Case 1: $x \geq 0$ In this case, $|x| = x$, so the inequality becomes:

$x \geq 11$

This tells us that all non-negative numbers greater than or equal to 11 satisfy the inequality. Case 2: $x < 0$ In this case, $|x| = -x$, so the inequality becomes:

$-x \geq 11$

To solve for $x$, we multiply both sides by -1. Remember that when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign:

$(-1)(-x) \leq (-1)(11)$

This simplifies to:

$x \leq -11$

This tells us that all negative numbers less than or equal to -11 also satisfy the inequality. Combining the two cases, we find that the solution to the inequality $|x| \geq 11$ consists of all real numbers $x$ such that $x \geq 11$ or $x \leq -11$. This means that any number that is 11 or more units away from zero on the number line will satisfy the inequality. This step-by-step breakdown ensures that the logic and process are clear, making it easier to understand how to solve inequalities involving absolute values.

After solving the inequality $6|x| \geq 66$, we need to express the solution in a clear and concise manner. We found that the solution consists of all real numbers $x$ such that $x \geq 11$ or $x \leq -11$. There are several ways to represent this solution. One common way is to use interval notation. In interval notation, we represent a range of numbers using brackets and parentheses. A square bracket [ or ] indicates that the endpoint is included in the interval, while a parenthesis ( or ) indicates that the endpoint is not included. In our case, $x \geq 11$ means that $x$ is in the interval $[11, \infty)$, where $\infty$ represents infinity. The square bracket on the left indicates that 11 is included in the interval, and the parenthesis on the right indicates that infinity is not a specific number but rather a concept of unboundedness. Similarly, $x \leq -11$ means that $x$ is in the interval $(-\infty, -11]$, where $-\infty$ represents negative infinity. The parenthesis on the left indicates that negative infinity is not a specific number, and the square bracket on the right indicates that -11 is included in the interval. To represent the entire solution, we combine these two intervals using the union symbol $\cup$. The union of two sets includes all elements that are in either set. Therefore, the solution to the inequality $6|x| \geq 66$ in interval notation is:

$(-\infty, -11] \cup [11, \infty)$

This notation clearly shows the range of values that satisfy the inequality. Another way to express the solution is using set-builder notation. In set-builder notation, we define a set by specifying the condition that its elements must satisfy. In our case, the solution can be written as:

${x \in \mathbb{R} \mid x \leq -11 \text{ or } x \geq 11}$

This is read as "the set of all real numbers $x$ such that $x$ is less than or equal to -11 or $x$ is greater than or equal to 11." Both interval notation and set-builder notation are standard ways to express solutions to inequalities in mathematics. The choice of which notation to use often depends on the context and personal preference. However, it is important to understand both notations to effectively communicate mathematical ideas.

Visualizing the solution to the inequality $6|x| \geq 66$ graphically can provide a deeper understanding of the result. We know that the solution consists of all real numbers $x$ such that $x \leq -11$ or $x \geq 11$. This means that on the number line, the solution includes all points to the left of and including -11, and all points to the right of and including 11. To represent this graphically, we draw a number line and mark the points -11 and 11. Since the inequality includes the points -11 and 11 (because of the "equal to" part of the inequality), we use closed circles or brackets at these points to indicate that they are part of the solution. Then, we shade the regions of the number line that correspond to the solution. This means shading the region to the left of -11 (representing $x \leq -11$) and the region to the right of 11 (representing $x \geq 11$). The unshaded region between -11 and 11 represents the values of $x$ that do not satisfy the inequality. This visual representation clearly shows the two separate intervals that make up the solution. The graphical representation is a powerful tool for understanding inequalities, especially those involving absolute values. It provides a visual confirmation of the algebraic solution and helps to reinforce the concept of a solution set. By seeing the solution on a number line, it becomes easier to grasp the range of values that satisfy the inequality and to understand the relationship between the inequality and its solution. This graphical approach is particularly useful for students who are visual learners, as it provides an additional way to process and understand the mathematical concepts involved.

In conclusion, we have successfully solved the inequality $6|x| \geq 66$ and explored various ways to express the solution. We began by understanding the concept of absolute value and its importance in solving such inequalities. We then proceeded with a step-by-step algebraic solution, isolating the absolute value term and considering the two cases arising from the definition of absolute value. This led us to the solution $x \leq -11$ or $x \geq 11$. Next, we discussed how to express the solution using interval notation and set-builder notation, which are standard ways to represent solution sets in mathematics. We found that the solution in interval notation is $(-\infty, -11] \cup [11, \infty)$, and in set-builder notation, it is ${x \in \mathbb{R} \mid x \leq -11 \text{ or } x \geq 11}$. Finally, we visualized the solution graphically on a number line, which provided a clear and intuitive understanding of the solution set. This comprehensive approach, covering algebraic, notational, and graphical representations, ensures a thorough understanding of the problem and its solution. Solving inequalities involving absolute values is a fundamental skill in mathematics, and the techniques discussed in this article can be applied to a wide range of similar problems. By mastering these techniques, students can build a strong foundation for more advanced mathematical concepts. The ability to solve inequalities is not only important in mathematics but also in various real-world applications, such as optimization problems, data analysis, and engineering. Therefore, a solid understanding of these concepts is invaluable for anyone pursuing a career in STEM fields or any field that requires quantitative reasoning.