Solve |x-4| < 3: A Step-by-Step Guide

Hey guys! Let's tackle an absolute value inequality problem today. We're going to break down the solution to $|x-4| < 3$ step by step, making sure everyone understands the logic and reasoning behind each move. Inequalities can seem tricky, especially when absolute values are involved, but don't worry, we'll make it crystal clear.

Understanding Absolute Value Inequalities

Before we dive into the specific problem, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero. So, $|5| = 5$ and $|-5| = 5$. This is crucial to understanding absolute value inequalities. When we see $|x-4| < 3$, it means the distance between x and 4 is less than 3.

Key Concepts and Properties

The main keyword here is absolute value inequality. Remember that absolute value represents the distance from zero. When solving inequalities involving absolute values, we need to consider two cases: the expression inside the absolute value can be positive or negative. This is because both a positive and a negative number within a certain distance from zero will satisfy the inequality. For the inequality $|x| < a$ (where a is a positive number), this translates into two separate inequalities: $-a < x < a$. This concept is the bedrock of solving the problem at hand, $|x-4| < 3$.

To properly grasp this, think of a number line. If $|x| < 3$, that means x can be any number between -3 and 3 (not including -3 and 3). The same principle applies when we have an expression inside the absolute value, like in our problem. We’re not just looking at x; we’re looking at the distance between x and 4.

Breaking Down |x-4| < 3

So, how do we apply this to $|x-4| < 3$? We need to split it into two separate inequalities. This is where the magic happens, guys! The absolute value inequality $|x-4| < 3$ is equivalent to the compound inequality $-3 < x-4 < 3$. This is absolutely essential to remember.

Let's break down why this works. The expression $|x-4|$ represents the distance between x and 4. The inequality $|x-4| < 3$ says this distance must be less than 3. This means x must be within 3 units of 4, in either direction. On a number line, this includes all numbers between 1 and 7.

Therefore, we can express this as two inequalities:

1. $x - 4 < 3$ (x is less than 3 units away from 4 on the right side)
2. $-(x - 4) < 3$ (x is less than 3 units away from 4 on the left side)

However, it’s much cleaner and more direct to write it as the compound inequality $-3 < x - 4 < 3$. This elegantly captures both conditions in a single statement. Understanding this equivalence is key to solving any absolute value inequality problem you might encounter.

Step-by-Step Solution

Now that we understand the core concept, let's solve the inequality $|x-4| < 3$ step-by-step. We've already established that this is equivalent to the compound inequality $-3 < x - 4 < 3$. Our goal is to isolate x in the middle.

Isolating x

The way we isolate x in a compound inequality is by performing the same operation on all three parts (left side, middle, and right side). In this case, we need to get rid of the -4 that’s being subtracted from x. The opposite of subtraction is addition, so we'll add 4 to all three parts of the inequality.

Starting with $-3 < x - 4 < 3$, we add 4 to each part:

$-3 + 4 < x - 4 + 4 < 3 + 4$

This simplifies to:

$1 < x < 7$

Interpreting the Solution

So, the solution to the inequality $|x-4| < 3$ is $1 < x < 7$. What does this mean? It means that x can be any number between 1 and 7, but it cannot be equal to 1 or 7. We often represent this solution graphically on a number line using an open interval. On the number line, we would draw a line segment between 1 and 7, with open circles at 1 and 7 to indicate that these endpoints are not included in the solution.

This inequality represents all real numbers strictly between 1 and 7. If x were equal to 1 or 7, the distance between x and 4 would be exactly 3, not less than 3. If x were less than 1 or greater than 7, the distance between x and 4 would be greater than 3, violating the original inequality. Therefore, the solution set consists of all real numbers strictly between 1 and 7, making $1 < x < 7$ the correct solution. It’s crucial to remember that we're looking for values of x that make the distance between x and 4 less than 3, not equal to 3.

Checking the Solution

It’s always a good idea to check your solution to make sure it's correct. We can do this by picking a number within the solution range ($1 < x < 7$) and plugging it back into the original inequality. Let's pick x = 4 (since 4 is nicely between 1 and 7) and substitute it into $|x-4| < 3$:

$|4 - 4| < 3$

$|0| < 3$

$0 < 3$

This is true! So, our solution is likely correct. We can also test a number outside the range, such as x = 0. Substituting this into the original inequality gives:

$|0 - 4| < 3$

$|-4| < 3$

$4 < 3$

This is false, confirming that numbers outside the range $1 < x < 7$ do not satisfy the original inequality. This step of verifying our solution using test values is an invaluable check to guarantee the accuracy of our response.

Analyzing the Answer Choices

Now let's look at the answer choices provided and see which one matches our solution, $1 < x < 7$.

A. $-7$: This is a single number, not a range, so it's incorrect.

B. $1$: This is also a single number, and it's not part of our solution (remember, x must be strictly greater than 1).

C. $x < -7$ or $x < -1$: This represents two intervals that are not part of our solution. Our solution is between 1 and 7, not less than -7 or -1.

D. $1 < x < 7$: This is exactly what we found! This represents the interval of numbers between 1 and 7, excluding 1 and 7.

Therefore, the correct answer is the inequality $1 < x < 7$.

Common Mistakes to Avoid

When dealing with absolute value inequalities, there are a few common mistakes students often make. Let's highlight some of these so you can avoid them:

1. Forgetting to split the inequality into two cases: This is the biggest mistake. Remember, $|x-4| < 3$ translates to two inequalities: $x-4 < 3$ and $-(x-4) < 3$ which simplifies to $-3 < x-4$. Failing to consider both the positive and negative possibilities inside the absolute value will lead to an incorrect solution. It’s imperative to always split the absolute value inequality into its equivalent compound inequality.

2. Incorrectly applying the inequality signs: When dealing with the negative case, remember to flip the inequality sign if you multiply or divide by a negative number. While we avoided multiplying by a negative in our solution by writing the compound inequality $-3 < x - 4 < 3$, this is a critical point to remember for other absolute value problems.

3. Misinterpreting the solution set: Make sure you understand what the solution set represents. For $1 < x < 7$, it means all numbers strictly between 1 and 7. Don't include 1 and 7 in your solution. Visualizing the solution on a number line can be extremely helpful in preventing this error. This visual representation makes it unambiguously clear what numbers are included and excluded from the solution.

4. Not checking the solution: Always check your solution by plugging in a value from your solution range and a value outside your range into the original inequality. This helps catch any algebraic errors you might have made. This simple step is an excellent way to verify that your answer is indeed correct.

Conclusion

So, there you have it! We've solved the inequality $|x-4| < 3$ and found that the solution is $1 < x < 7$. The key takeaway here is understanding how to break down absolute value inequalities into compound inequalities. Remember to consider both the positive and negative cases of the expression inside the absolute value. By practicing these steps and avoiding common pitfalls, you’ll be solving absolute value inequalities like a pro!

If you have any questions or want to try more examples, let me know in the comments below. Keep practicing, and you'll master these concepts in no time!