Solving Absolute Value Inequalities

Hey guys, ever found yourself staring at an absolute value inequality and thinking, "What in the world does this even mean?" You're not alone! Today, we're going to break down one of those tricky problems: $|x+1|+12>9$. We'll explore how to solve it and discuss why the answer might surprise you. Get ready to level up your math game because we're diving deep into the fascinating world of inequalities!

Understanding Absolute Value

Before we jump into solving, let's quickly refresh what absolute value is all about. Remember, the absolute value of a number is its distance from zero on the number line. It's always a non-negative value. So, $|5| = 5$ and $|-5| = 5$. When we see an absolute value in an inequality, like in our problem $|x+1|+12>9$, it means we're dealing with a range of values for the expression inside the absolute value bars.

In our specific inequality, $|x+1|+12>9$, the first thing we want to do is isolate the absolute value term. Think of it like untangling a knot – we need to get the tricky part by itself so we can figure out what's going on. To do this, we'll subtract 12 from both sides of the inequality. This gives us: $|x+1| > 9 - 12$, which simplifies to $|x+1| > -3$.

Now, let's pause and really think about this. We have the absolute value of some expression, $|x+1|$, and we're saying it's greater than -3. What do we know about absolute values? They are always greater than or equal to zero. They represent a distance, and distance can't be negative, right? So, no matter what value $x$ takes, $|x+1|$ will always be 0 or a positive number. Since any non-negative number is inherently greater than any negative number, like -3, this inequality $|x+1| > -3$ is true for every single possible value of $x$. Mind-blowing, huh? This is why understanding the fundamental properties of absolute value is so crucial when tackling these problems. It saves us a ton of confusion down the line and helps us see the elegant simplicity that often lies beneath complex-looking equations.

Solving the Inequality: Step-by-Step

Alright, let's walk through the steps to officially solve $|x+1|+12>9$. Our main goal here is to isolate the absolute value expression, $|x+1|$, so we can understand the conditions under which the inequality holds true. This is a standard strategy for dealing with absolute value equations and inequalities, and it's super effective.

Step 1: Isolate the absolute value.

We start with the original inequality:

$|x+1|+12>9$

To get $|x+1|$ by itself on one side, we need to get rid of that '+12'. The opposite of adding 12 is subtracting 12, so we'll do that to both sides of the inequality to keep things balanced:

$|x+1|+12 - 12 > 9 - 12$

This simplifies to:

$|x+1| > -3$

Step 2: Analyze the resulting inequality.

Now we have $|x+1| > -3$. This is where a key property of absolute value comes into play. Remember, the absolute value of any number or expression is always non-negative. That means $|x+1|$ will always be greater than or equal to 0, no matter what real number $x$ happens to be.

Think about it: if $x=5$, then $|x+1| = |5+1| = |6| = 6$, which is greater than -3. If $x=-1$, then $|x+1| = |-1+1| = |0| = 0$, which is greater than -3. If $x=-10$, then $|x+1| = |-10+1| = |-9| = 9$, which is also greater than -3.

Since $|x+1|$ is guaranteed to be non-negative (i.e., 0 or positive), it will always be greater than any negative number. In this case, -3 is negative. Therefore, the inequality $|x+1| > -3$ is true for all real numbers $x$.

Step 3: Determine the solution set.

Because the inequality is true for every single real number, the solution set is not limited to a specific range or a few values. Instead, it encompasses the entirety of the number line. This means our solution set is all real numbers.

So, when we look at the options provided:

A. The solution set is $\varnothing$. (This means there are no solutions, which is incorrect.) B. The solution set is $-4$ (This is just a single value, not a set, and doesn't satisfy the inequality for all x). C. The solution set is $x<-4$ OR $x>2$. (This represents a specific range, not all real numbers). D. The solution set is all real numbers. (This matches our findings!)

Therefore, the correct answer is D. It's awesome how sometimes the most complex-looking problems have the most straightforward answers once you break them down!

Why "All Real Numbers" Isn't Always Obvious

Okay, so we figured out that $|x+1|+12>9$ is true for all real numbers, which corresponds to option D. But let's be real, guys, sometimes seeing "all real numbers" as the answer can feel a bit anticlimactic or even confusing. Why? Because usually, when we solve inequalities, we expect to get a specific range of values, like $x > 5$ or $x < -2$ or maybe even a combination like $x < -4$ OR $x > 2$. So, when the solution is everything, it can make you second-guess yourself. "Did I do this right?" "Is there a trick I missed?"

The key takeaway here, and it's a super important one, is to always pay attention to the fundamental properties of the mathematical concepts you're working with. In this case, it's the absolute value. Remember, absolute value is never negative. It's a distance. It can be zero, or it can be positive, but it cannot dip into the negative territory.

When our inequality simplified to $|x+1| > -3$, we were essentially asking: "When is a non-negative number greater than a negative number?" The answer is always. Think about it on a number line. Any number that's zero or to the right of zero (non-negative) is always going to be to the right of any negative number. $-3$ is way over on the left side of the number line, and $|x+1|$ is always somewhere on the right side (or at zero). So, the condition is always met.

This is why understanding the nature of absolute value is so critical. If the inequality had been, say, $|x+1| < -3$, then the answer would have been the empty set ($\varnothing$, option A), because a non-negative number can never be less than a negative number. If it had been $|x+1| > 3$, then we would have split it into two separate inequalities: $x+1 > 3$ (which gives $x>2$) OR $x+1 < -3$ (which gives $x<-4$), leading to option C. See how the number on the right side of the inequality sign dramatically changes the outcome?

So, the next time you encounter an absolute value inequality, don't just blindly apply rules. Take a moment to consider what the absolute value part means. Is it possible for it to be less than a negative number? Is it always going to be greater than a negative number? This little bit of conceptual thinking can save you a lot of time and prevent those "oh no, what did I do wrong?" moments. It's all about building that strong mathematical intuition, guys!

Exploring the Options: Why Others Don't Fit

Let's quickly chat about why the other options (A, B, and C) are definitely not the correct solutions for $|x+1|+12>9$. It's super helpful to be able to eliminate incorrect answers, as it builds confidence in the one you choose.

• Option A: The solution set is $\varnothing$. This option suggests that there are no values of $x$ that satisfy the inequality. As we discovered, after simplifying the inequality to $|x+1| > -3$, we found that every real number works. The absolute value of something is always greater than or equal to zero, and any non-negative number is always greater than -3. So, there are definitely solutions, meaning the empty set is incorrect.

• Option B: The solution set is $-4$. This implies that only the single value $x = -4$ makes the inequality true. Let's test it out just to be sure. If $x = -4$, then $|x+1|+12 = |-4+1|+12 = |-3|+12 = 3+12 = 15$. Is $15 > 9$? Yes, it is. So, $x=-4$ is a solution. However, the question asks for the solution set, and we've already established that many other numbers work too (like $x=0$, which gives $|0+1|+12 = 1+12 = 13 > 9$). Therefore, a single number cannot be the entire solution set.

• Option C: The solution set is $x<-4$ OR $x>2$. This option represents a specific range of solutions, excluding the numbers between -4 and 2 (inclusive). This kind of solution typically arises when you have an inequality like $|x+1| > k$ where $k$ is a positive number. For example, if we had $|x+1| > 3$, we would split it into $x+1 > 3$ (giving $x>2$) and $x+1 < -3$ (giving $x<-4$). Since our simplified inequality was $|x+1| > -3$, which involves a negative number, we don't get this kind of split solution. All numbers satisfy it, not just those outside a certain range.

By understanding why these options are incorrect, we reinforce our understanding of the principles of absolute value inequalities and why D. The solution set is all real numbers is the only logical conclusion for the inequality $|x+1|+12>9$. Keep practicing, and you'll get the hang of spotting these patterns in no time!