Solving $-19 \leq 6x - 1 \leq 17$ A Step-by-Step Guide

In this article, we will delve into the process of solving a compound inequality. Compound inequalities, as the name suggests, involve two or more inequalities combined into a single statement. These inequalities often involve expressions that must satisfy multiple conditions simultaneously. Specifically, we will address the compound inequality $-19 \leq 6x - 1 \leq 17$. This type of problem is a fundamental concept in algebra and is crucial for understanding more advanced mathematical topics. The goal here is to find the set of all x values that satisfy both inequalities concurrently. This involves isolating the variable x by performing algebraic operations on all parts of the inequality. This step-by-step guide will break down the process into manageable steps, ensuring a clear understanding of each stage.

To successfully tackle the inequality $-19 \leq 6x - 1 \leq 17$, it is essential to first grasp the concept of compound inequalities. A compound inequality is essentially a combination of two or more inequalities connected by either "and" or "or." In our case, the given inequality is a concise way of writing two separate inequalities: $-19 \leq 6x - 1$ and $6x - 1 \leq 17$. The word "and" here signifies that both inequalities must be true at the same time. This means that the solution set will consist of all x values that satisfy both inequalities simultaneously. Understanding this "and" condition is crucial because it dictates how we approach solving the compound inequality. We need to manipulate the expression to isolate x while ensuring that the changes we make apply equally to all parts of the inequality. This ensures that we maintain the integrity of the compound statement and arrive at the correct solution set. Solving compound inequalities is a cornerstone of algebraic manipulation and is frequently encountered in various mathematical and real-world applications.

Now, let's embark on the step-by-step solution of the compound inequality $-19 \leq 6x - 1 \leq 17$. Our primary objective is to isolate x in the middle of the inequality. To achieve this, we will perform a series of algebraic operations, ensuring that each operation is applied uniformly to all three parts of the inequality. This is crucial for maintaining the balance and validity of the compound statement. First, we'll add 1 to all parts of the inequality. This operation cancels out the -1 term in the middle, bringing us closer to isolating x. The inequality then transforms to $-19 + 1 \leq 6x - 1 + 1 \leq 17 + 1$, which simplifies to $-18 \leq 6x \leq 18$. Next, we need to eliminate the coefficient 6 that is multiplying x. To do this, we will divide all parts of the inequality by 6. This gives us $\frac{-18}{6} \leq \frac{6x}{6} \leq \frac{18}{6}$, which simplifies to $-3 \leq x \leq 3$. This final inequality tells us that x must be greater than or equal to -3 and less than or equal to 3. We have now successfully isolated x and found the solution set for the compound inequality.

Step 1: Add 1 to All Parts

The first crucial step in solving the compound inequality $-19 \leq 6x - 1 \leq 17$ is to isolate the term containing x. This involves eliminating the constant term that is being added or subtracted from the variable term. In this case, we have a -1 term. To counteract this, we add 1 to all three parts of the inequality. This maintains the balance of the inequality and moves us closer to isolating x. The operation can be written as: $-19 + 1 \leq 6x - 1 + 1 \leq 17 + 1$. Performing the addition, we simplify the inequality to $-18 \leq 6x \leq 18$. This new form of the inequality is much easier to work with because the constant term has been eliminated from the middle part, bringing us closer to isolating x. This step is a fundamental technique in solving inequalities and is analogous to adding the same value to both sides of an equation. By applying this addition across all parts of the compound inequality, we preserve the relationship between the expressions and set the stage for the next steps in the solution process. This methodical approach ensures that we arrive at the correct solution set while maintaining the mathematical integrity of the problem.

Step 2: Divide All Parts by 6

Following the addition step, we now have the inequality $-18 \leq 6x \leq 18$. The next logical step is to isolate x completely by removing its coefficient. In this case, x is multiplied by 6. To undo this multiplication, we divide all three parts of the inequality by 6. This operation is valid as long as we divide by a positive number, which is the case here. Dividing by a positive number does not change the direction of the inequality signs. The operation can be expressed as: $\frac{-18}{6} \leq \frac{6x}{6} \leq \frac{18}{6}$. Performing the division, we simplify each part of the inequality. $\frac{-18}{6}$ equals -3, $\frac{6x}{6}$ simplifies to x, and $\frac{18}{6}$ equals 3. Thus, the inequality becomes $-3 \leq x \leq 3$. This resulting inequality provides a clear and concise definition of the solution set for x. It states that x must be greater than or equal to -3 and simultaneously less than or equal to 3. This step is crucial because it effectively isolates x, revealing the range of values that satisfy the original compound inequality. By dividing all parts of the inequality by 6, we have successfully determined the bounds within which x must lie, providing the final piece of the solution.

After solving the compound inequality, it is crucial to accurately express the solution set. In our case, the solution is $-3 \leq x \leq 3$. This inequality can be represented in several ways, each offering a different perspective on the solution. Firstly, we can express the solution in interval notation. Interval notation uses brackets and parentheses to indicate the range of values that satisfy the inequality. A square bracket [ ] indicates that the endpoint is included in the solution, while a parenthesis ( ) indicates that the endpoint is excluded. Since our inequality includes -3 and 3 (because of the "less than or equal to" and "greater than or equal to" signs), we use square brackets. Therefore, the interval notation for the solution is $[-3, 3]$. This notation concisely represents all real numbers between -3 and 3, inclusive. Secondly, we can represent the solution graphically on a number line. This involves drawing a number line and marking the relevant interval. We would place closed circles (or brackets) at -3 and 3 to indicate that these points are included in the solution. Then, we would shade the region between -3 and 3 to represent all the values of x that satisfy the inequality. This graphical representation provides a visual understanding of the solution set. Finally, we can express the solution in set-builder notation. Set-builder notation describes the solution set using the properties that its elements must satisfy. For our solution, the set-builder notation would be { x | x is a real number and $-3 \leq x \leq 3$ }. This notation reads as “the set of all x such that x is a real number and x is between -3 and 3, inclusive.” Each of these representations – interval notation, graphical representation, and set-builder notation – provides a valuable way to communicate the solution to the compound inequality, catering to different preferences and contexts.

In conclusion, solving the compound inequality $-19 \leq 6x - 1 \leq 17$ involves a systematic approach of isolating the variable x. We achieved this by first adding 1 to all parts of the inequality and then dividing all parts by 6, resulting in the simplified inequality $-3 \leq x \leq 3$. This solution can be expressed in various forms, including interval notation $[-3, 3]$, a graphical representation on a number line, and set-builder notation { x | x is a real number and $-3 \leq x \leq 3$ }. Understanding how to solve compound inequalities is a fundamental skill in algebra, laying the groundwork for more complex mathematical problems. This process reinforces the importance of applying operations consistently across all parts of an inequality to maintain its balance and accuracy. Furthermore, the ability to express the solution in multiple forms highlights the versatility of mathematical language and the importance of choosing the representation that best suits the context. Mastering these techniques is not only crucial for academic success in mathematics but also for problem-solving in various real-world scenarios that involve constraints and conditions. The skills acquired in solving compound inequalities enhance logical reasoning and analytical abilities, which are valuable assets in a wide range of disciplines.