Solve |x-5|+2 < 20: A Step-by-Step Guide
Hey guys! Ever stumbled upon an absolute value inequality and felt a bit lost? Don't worry, we've all been there. Absolute values can seem tricky, but with a clear method, you can crack them easily. Today, we're diving into how to solve the inequality $|x-5|+2 < 20$. We'll break it down step-by-step, making sure you understand not just the how, but also the why behind each move. So, buckle up and let's get started!
Understanding Absolute Value Inequalities
Before we jump into solving our specific problem, let's quickly recap what absolute value means. Absolute value, at its core, represents the distance a number is from zero on the number line. It's always non-negative. For example, both |3| and |-3| equal 3 because both 3 and -3 are three units away from zero. When we deal with inequalities involving absolute values, like the one we're tackling today, we're essentially looking for a range of numbers that satisfy a certain distance condition.
So, when you see an inequality like $|x-5|+2 < 20$, think of it as asking: "For what values of x is the distance between x and 5 (after some transformations) less than 20?" This understanding is crucial because it helps us visualize the solution on a number line and avoid common pitfalls. The key idea here is that absolute value expressions often lead to two separate cases that we need to consider. We'll see how this plays out as we solve our inequality. We'll explore this concept thoroughly throughout this guide, ensuring you have a solid grasp of the fundamental principles behind absolute value inequalities. Remember, mastering the basics is crucial for tackling more complex problems later on. Let's move on to the first step in solving our inequality: isolating the absolute value expression.
Step 1: Isolate the Absolute Value
Okay, our first order of business is to get the absolute value expression, $|x-5|$, all by itself on one side of the inequality. This is like clearing the stage for the main act! Right now, we have $|x-5|+2 < 20$. That pesky '+2' is cramping our style. To get rid of it, we'll use the good old inverse operation trick. We'll subtract 2 from both sides of the inequality. Remember, whatever we do to one side, we must do to the other to keep the balance. So, subtracting 2 from both sides gives us:
This simplifies beautifully to:
Great! Now we've successfully isolated the absolute value expression. This is a crucial step because it sets us up to deal with the two possible cases that arise from the absolute value. Think of it like this: we've peeled away the outer layers and now we can see the core of the problem. By isolating the absolute value, we've made it much clearer what we need to address. This step is not just about following a rule; it's about simplifying the problem so we can apply the definition of absolute value effectively. Next up, we'll split this into two separate inequalities, each representing a different scenario within the absolute value.
Step 2: Split into Two Cases
Now for the fun part! Remember how we talked about absolute value representing distance from zero? This is where that understanding really shines. The inequality $|x-5| < 18$ tells us that the distance between x and 5 is less than 18. But there are two ways this can happen. x - 5 could be a positive number less than 18, or it could be a negative number whose absolute value is less than 18. This leads us to our two cases:
Case 1: The expression inside the absolute value is positive or zero. In this case, we can simply drop the absolute value bars:
Case 2: The expression inside the absolute value is negative. In this case, we need to change the sign of the expression inside the absolute value and flip the inequality sign:
Notice that we're not just changing the sign of one term; we're changing the sign of the entire expression inside the absolute value. This is super important! We're essentially saying that if the original expression is negative, its negative counterpart must also be less than 18. Now we have two simple inequalities to solve, each representing a piece of the puzzle. Solving these inequalities will give us the range of x values that satisfy our original absolute value inequality. Let's tackle Case 1 first.
Step 3: Solve Case 1
Alright, let's tackle Case 1: $x - 5 < 18$. This one is pretty straightforward. Our goal is to isolate x, so we need to get rid of that '- 5'. Just like before, we'll use the inverse operation. To undo subtraction, we'll add! We add 5 to both sides of the inequality:
This simplifies to:
Boom! We've solved for x in Case 1. This tells us that all values of x less than 23 satisfy the first condition of our absolute value inequality. But we're not done yet! Remember, we had two cases to consider. So, we need to keep this solution in mind and move on to Case 2. It's like we've found one piece of the puzzle, but we need the other piece to see the whole picture. Solving Case 1 was a crucial step, but it only gives us half the story. Now, let's dive into Case 2 and see what other restrictions x might have.
Step 4: Solve Case 2
Time to tackle Case 2: $-(x - 5) < 18$. This one requires a little more finesse. First, we need to deal with that negative sign outside the parentheses. We can do this by distributing the negative sign across the terms inside the parentheses:
Now, we want to isolate x. Let's start by subtracting 5 from both sides:
This simplifies to:
But wait! We're not quite there yet. We have -x on the left side, and we want x. To get rid of the negative sign, we can multiply (or divide) both sides by -1. But here's the crucial part: when we multiply or divide an inequality by a negative number, we have to flip the inequality sign! So, multiplying both sides by -1 gives us:
There we go! We've solved for x in Case 2. This tells us that all values of x greater than -13 also satisfy our absolute value inequality. We're getting closer to the final solution. We've solved both cases, and now we need to combine them to get the complete picture. This involves understanding how the solutions from each case fit together. Let's move on to the final step: combining our solutions.
Step 5: Combine the Solutions
We've done the hard work! We solved both cases of our absolute value inequality and found that x < 23 and x > -13. Now, we need to put these two pieces together to get the complete solution. Think of it like this: we have two conditions that x must satisfy. It has to be less than 23, and it has to be greater than -13.
To visualize this, you can imagine a number line. We have a point at -13 and a point at 23. x has to be to the right of -13 (but not including -13) and to the left of 23 (but not including 23). This means x is trapped between -13 and 23.
We can write this combined solution as a compound inequality:
This is the final solution to our problem! It tells us that any value of x between -13 and 23 (not including -13 and 23 themselves) will satisfy the original inequality, $|x-5|+2 < 20$. We've successfully navigated the twists and turns of absolute value inequalities and arrived at our answer. Congrats! We've successfully found the solution set for the given absolute value inequality. Let's recap the steps we took and solidify our understanding.
Final Answer and Recap
Phew! We made it. The solution to the inequality $|x-5|+2 < 20$ is $-13 < x < 23$. That corresponds to option B.
Let's quickly recap the steps we took to get there:
- Isolate the absolute value: We subtracted 2 from both sides to get $|x-5| < 18$.
- Split into two cases: We considered both the positive and negative possibilities inside the absolute value, leading to $x - 5 < 18$ and $-(x - 5) < 18$.
- Solve Case 1: We solved $x - 5 < 18$ to get $x < 23$.
- Solve Case 2: We solved $-(x - 5) < 18$ to get $x > -13$.
- Combine the solutions: We combined $x < 23$ and $x > -13$ to get $-13 < x < 23$.
Remember, the key to solving absolute value inequalities is to understand the concept of distance from zero and to consider both positive and negative possibilities. By following these steps, you can confidently tackle any absolute value inequality that comes your way. Keep practicing, and you'll become an absolute value pro in no time! Remember, mathematics is a journey, and each problem solved is a step forward. Keep practicing and exploring, and you'll be amazed at what you can achieve. Until next time, keep those problem-solving skills sharp!
