Solving |x-1| ≤ 6: Interval Notation Explained

Hey guys! Today, we're diving into solving an absolute value inequality and expressing the solution in interval notation. Specifically, we're tackling the inequality |x-1| ≤ 6. This type of problem often appears in math courses, and understanding how to solve it is super useful. So, let's break it down step by step to make sure we've got a solid grasp on the concept.

Understanding Absolute Value Inequalities

Before we jump into the specifics, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero. So, |x| represents the distance of x from zero, regardless of whether x is positive or negative. When we deal with absolute value inequalities like |x-1| ≤ 6, we're essentially saying that the distance between x-1 and zero is less than or equal to 6. This gives us two possible scenarios to consider:

1. x-1 is within 6 units to the right of zero.
2. x-1 is within 6 units to the left of zero.

Key Concept: An absolute value inequality of the form |ax + b| ≤ c can be rewritten as two separate inequalities: -c ≤ ax + b ≤ c. This is because ax + b can be between -c and c (inclusive) while still satisfying the absolute value condition. Understanding this fundamental concept is crucial for correctly solving these types of problems. It allows us to transform a single absolute value inequality into a compound inequality that we can then solve using standard algebraic techniques.

Remember, the goal is to isolate x and find the range of values that satisfy the original inequality. Keeping this in mind will help you navigate through the steps and arrive at the correct solution. So, let's move on and apply this concept to our specific problem.

Breaking Down |x-1| ≤ 6

Okay, now let's apply this to our problem: |x-1| ≤ 6. According to our key concept, we can rewrite this absolute value inequality as a compound inequality:

-6 ≤ x - 1 ≤ 6

This compound inequality tells us that x - 1 must be greater than or equal to -6 AND less than or equal to 6. Think of it as x - 1 being trapped between -6 and 6, inclusive. Now, our goal is to isolate x in the middle. To do this, we'll perform the same operation on all three parts of the inequality.

In this case, we need to get rid of the -1 that's being subtracted from x. To do this, we'll add 1 to all parts of the inequality:

-6 + 1 ≤ x - 1 + 1 ≤ 6 + 1

Simplifying this, we get:

-5 ≤ x ≤ 7

Alright! We've successfully isolated x. This inequality tells us that x must be greater than or equal to -5 AND less than or equal to 7. In other words, x can be any number between -5 and 7, including -5 and 7 themselves. It's super important to remember that the "or equal to" part of the inequality (≤) means that the endpoints are included in the solution. This will be crucial when we express our answer in interval notation.

So, now that we know the range of values for x, let's express this solution in interval notation.

Expressing the Solution in Interval Notation

Interval notation is a way of writing down a set of numbers using intervals. It uses brackets and parentheses to indicate whether the endpoints of the interval are included or excluded. A square bracket [ or ] indicates that the endpoint is included in the set, while a parenthesis ( or ) indicates that the endpoint is not included. Since our inequality is -5 ≤ x ≤ 7, this means that x can be -5, 7, or any number in between.

Therefore, we use square brackets to include both -5 and 7 in our interval. The interval notation for -5 ≤ x ≤ 7 is:

[-5, 7]

This notation tells us that the solution set includes all real numbers from -5 to 7, including -5 and 7. The left bracket [ indicates that -5 is included, and the right bracket ] indicates that 7 is included. If the endpoints were not included (for example, if the inequality was -5 < x < 7), we would use parentheses instead: (-5, 7). So, make sure you pay close attention to whether the endpoints are included or excluded when writing your solution in interval notation.

Why the Other Options are Incorrect

Now, let's quickly look at why the other answer options are incorrect:

• A. (-\infty,-5] or [7, \infty): This represents all numbers less than or equal to -5 or greater than or equal to 7. This is the solution to |x-1| ≥ 6, not |x-1| ≤ 6. This option describes values outside the range we need.
• B. (-\infty,-5) or (7, \infty): Similar to option A, this represents numbers less than -5 or greater than 7, but excluding -5 and 7. Again, this is the solution to |x-1| > 6. This option also describes values outside the range we need and excludes the endpoints.
• D. (-5,7): This represents all numbers between -5 and 7, excluding -5 and 7. This would be the solution if our original inequality was |x-1| < 6. This option is close, but it misses the crucial detail that -5 and 7 are included in the solution because of the "less than or equal to" sign.

Understanding why these options are wrong helps reinforce why [-5, 7] is the correct answer. It's all about paying attention to the details of the inequality and understanding what the symbols mean.

Conclusion

So, the solution to the absolute value inequality |x-1| ≤ 6 in interval notation is [-5, 7]. Remember, solving absolute value inequalities involves breaking them down into compound inequalities, isolating x, and then expressing the solution in interval notation. Pay close attention to whether the endpoints are included or excluded to ensure you use the correct brackets or parentheses.

By understanding these steps and practicing, you'll become a pro at solving absolute value inequalities! Keep practicing, and you'll nail it every time. Good luck, and happy problem-solving!