Solving $|2x| < 18$: Your Step-by-Step Guide

When you first see an absolute value inequality like $|2x| < 18$, it might seem a little intimidating. But don't worry! We're going to break it down step-by-step, making it super clear and easy to understand. This inequality essentially asks: "What numbers, when multiplied by 2, result in a value that is less than 18 units away from zero?" The absolute value symbol, $| |$, means we're only concerned with the distance from zero, not the direction. So, a number like -5 has an absolute value of 5, and a number like 5 also has an absolute value of 5. For $|2x| < 18$, we're looking for values of $x$ where the expression $2x$ is between -18 and 18. To solve this, we can rewrite the absolute value inequality as a compound inequality. This means we need to consider two possibilities: either $2x$ is positive and less than 18, or $2x$ is negative and greater than -18. Both of these conditions can be combined into a single statement: $-18 < 2x < 18$. This is the core of solving most absolute value inequalities of the form $|ax+b| < c$. You want to set the expression inside the absolute value to be between $-c$ and $c$. In our case, $a=2$, $b=0$, and $c=18$. Once we have this compound inequality, the next step is to isolate $x$. To do this, we perform the same operation on all three parts of the inequality. Since $x$ is being multiplied by 2, we will divide all parts of the inequality by 2. Dividing $-18$ by 2 gives us $-9$. Dividing $2x$ by 2 gives us $x$. And dividing $18$ by 2 gives us $9$. So, our inequality becomes $-9 < x < 9$. This means that $x$ can be any number strictly between -9 and 9. It cannot be -9 or 9 themselves because the original inequality was "less than" ($<$) and not "less than or equal to" ($\le$). This range of numbers is often represented in interval notation. An open interval, indicated by parentheses, means the endpoints are not included. Therefore, the solution set for $|2x| < 18$ is the interval $(-9, 9)$. This interval includes all real numbers greater than -9 and less than 9. The other options presented, such as $[-9,9]$, $(-\infty, 9) \cup (9, \infty)$, and $(-\infty, 9)$, do not accurately represent the solution to this specific absolute value inequality.

Understanding Absolute Value Inequalities

Let's dive a bit deeper into why we can split $|2x| < 18$ into $-18 < 2x < 18$. The definition of absolute value is crucial here. For any real number $y$, $|y|$ is its distance from zero on the number line. This means $|y| = y$ if $y \ge 0$, and $|y| = -y$ if $y < 0$. When we have an inequality like $|A| < c$ (where $c$ is a positive number), it signifies that the quantity $A$ is less than $c$ units away from zero. On a number line, this translates to $A$ being located between $-c$ and $c$. Think about it: if a number is less than 5 units away from zero, it can be at 4, 3, 2, 1, 0, -1, -2, -3, or -4. It cannot be at 5 or -5, nor can it be at 6 or -6. Thus, the condition $|A| < c$ is equivalent to $-c < A < c$. Applying this to our specific problem, $A$ is $2x$ and $c$ is $18$. Therefore, $|2x| < 18$ is indeed equivalent to $-18 < 2x < 18$. This compound inequality is a fundamental concept in algebra. It allows us to transform an absolute value problem into a more manageable linear inequality problem. The power of this transformation lies in its universality for this type of inequality. Any time you encounter an absolute value inequality of the form $|expression| < positive number$, you can apply this rule. It simplifies the problem significantly, reducing it to a matter of solving a basic three-part inequality. The goal then becomes isolating the variable, which in our case is $x$. We achieve this by performing inverse operations. Since $2x$ means $2$ multiplied by $x$, the inverse operation is division. We must apply this division by 2 to all parts of the compound inequality to maintain the balance and correctness of the relationships. This ensures that the range of values for $x$ that satisfy the original absolute value inequality remains accurate. The process of isolating $x$ is where we transition from understanding the abstract concept of absolute value to finding concrete numerical solutions. It's a bridge between theoretical definitions and practical algebraic manipulation. Mastering this step is key to confidently solving a wide array of absolute value problems, making complex-looking equations much more approachable and solvable.

Isolating the Variable '$x$'

Now that we have established that $|2x| < 18$ is equivalent to the compound inequality $-18 < 2x < 18$, our next crucial step is to isolate the variable $x$. This means we want to get $x$ by itself in the middle part of the inequality. To do this, we need to undo the operations that are currently being applied to $x$. In the expression $2x$, $x$ is being multiplied by $2$. The inverse operation of multiplication is division. Therefore, to isolate $x$, we must divide the term $2x$ by $2$. However, it's absolutely vital to remember that when working with inequalities, any operation you perform on one part of the inequality must be performed on all parts to maintain the validity of the inequality. So, we need to divide all three sections of the compound inequality $-18 < 2x < 18$ by $2$. Let's do that systematically:

1. Left side: Divide $-18$ by $2$. $-18 \div 2 = -9$.
2. Middle section: Divide $2x$ by $2$. $2x \div 2 = x$.
3. Right side: Divide $18$ by $2$. $18 \div 2 = 9$.

After performing these divisions, our compound inequality transforms from $-18 < 2x < 18$ to $-9 < x < 9$. This new inequality tells us that the values of $x$ that satisfy the original absolute value inequality are all the numbers that are strictly greater than $-9$ and strictly less than $9$. The use of the greater than ($>$) and less than ($<$) signs, rather than greater than or equal to ($\ge$) or less than or equal to ($\le$), directly stems from the original inequality being $|2x| < 18$. The strict inequality symbol in the original problem means that the boundary values of $-9$ and $9$ are not included in the solution set. If the original problem had been $|2x| \le 18$, then our final inequality would have been $-9 \le x \le 9$, and the endpoints would have been included. The process of isolating the variable is a fundamental algebraic technique that applies to solving all sorts of equations and inequalities, not just absolute value ones. It relies on the principle of maintaining balance. Just like in a scale where you must add or remove equal weights from both sides to keep it balanced, in inequalities, you must perform identical operations on all parts to ensure the relationships (greater than, less than, etc.) remain true. This methodical approach ensures accuracy and prevents common errors that can arise from overlooking this crucial rule. By diligently applying this principle, we can confidently arrive at the correct solution set for our absolute value inequality.

Interpreting the Solution Set: $(-9, 9)$

We have successfully transformed the absolute value inequality $|2x| < 18$ into the compound inequality $-9 < x < 9$. Now, it's important to understand what this result actually means and how it relates to the given options. The inequality $-9 < x < 9$ specifies a range of values for $x$. It means that $x$ must be a number that is greater than -9 and simultaneously less than 9. This is a continuous range of numbers on the number line. When we talk about numbers being strictly greater than or strictly less than a certain value, we are referring to an open interval. In mathematics, open intervals are typically represented using parentheses. The notation $(a, b)$ signifies all numbers $x$ such that $a < x < b$. Conversely, a closed interval, denoted by $[a, b]$, represents all numbers $x$ such that $a \le x \le b$, meaning the endpoints $a$ and $b$ are included. Since our solution is $-9 < x < 9$, neither $-9$ nor $9$ are included in the solution set. This is because the original inequality used the "less than" symbol ($<$) and not the "less than or equal to" symbol ($\le$). Therefore, the correct way to express this solution set in interval notation is $(-9, 9)$. This notation clearly communicates that the interval includes all real numbers between $-9$ and $9$, excluding $-9$ and $9$ themselves.

Let's examine why the other options are incorrect:

• A. $[-9, 9]$: This represents all numbers $x$ such that $-9 \le x \le 9$. The square brackets indicate that the endpoints $-9$ and $9$ are included. This would be the correct answer if the original inequality was $|2x| \le 18$, but it's not.
• B. $(-\infty, 9) \cup (9, \infty)$: This represents all numbers $x$ that are either less than $9$ OR greater than $9$. This is the solution to an inequality like $|2x| > 18$. It excludes the numbers between $-9$ and $9$ (inclusive of $-9$ and $9$ if it were $\ge$), which is precisely the range we found. This is clearly not our solution.
• D. $(-\infty, 9)$: This represents all numbers $x$ such that $x < 9$. This inequality includes numbers like $-10, -100, -1000$, and so on, which would make $|2x|$ much larger than $18$. For example, if $x = -10$, then $|2x| = |-20| = 20$, which is not less than $18$. So, this option is also incorrect.

Thus, the only correct representation of the solution to $|2x| < 18$ is the open interval $(-9, 9)$. This is option C.

Conclusion

Solving the absolute value inequality $|2x| < 18$ boils down to understanding the fundamental property of absolute values: distance from zero. By translating the inequality into a compound inequality, $-18 < 2x < 18$, we transformed the problem into a more manageable form. The subsequent step of isolating $x$ by dividing all parts of the inequality by $2$ yielded the result $-9 < x < 9$. This final inequality represents all real numbers strictly between $-9$ and $9$. In interval notation, this is expressed as $(-9, 9)$, which corresponds to option C. It's essential to remember the difference between strict inequalities ($<$ or $>$) and non-strict inequalities ($\le$ or $\ge$), as this determines whether the endpoints are included in the solution set. Mastering these steps will equip you to tackle a wide variety of absolute value problems with confidence.

For more insights into absolute value inequalities and their solutions, you can explore resources like ** **Khan Academy's Absolute Value Equations and Inequalities section**.