Solving Absolute Value Inequalities: A Step-by-Step Guide

Hey guys! Today we're diving deep into the world of absolute value inequalities. You know, those tricky problems that can sometimes make your brain do a backflip? We're going to tackle a specific one: **solve for $z$ in the inequality $1-5|-4 z|

Write a compound inequality like $1 < x < 3$ or like $x < 1$ or $x > 3$. Use integers, proper fractions, or improper fractions in simplest form.** Sounds like a mouthful, right? But don't sweat it! We're going to break it down piece by piece, so by the end of this, you'll be an absolute value master. We'll make sure to cover all the bases, from understanding what an absolute value inequality even means to writing the final answer in the correct compound inequality format. So, grab your notebooks, maybe a comfy chair, and let's get this math party started! We promise to keep it fun, engaging, and super informative, just the way you like it here at Plastik Magazine.

Understanding Absolute Value Inequalities

Alright, let's kick things off by getting a solid grip on what absolute value inequalities actually are. Think of the absolute value, denoted by those vertical bars like $|x|$, as the distance of a number from zero on the number line. It's always a non-negative value. So, $|5| = 5$ and $|-5| = 5$. Pretty straightforward, right? Now, when we throw an inequality sign into the mix, like $|x| < 3$ or $|x| > 2$, we're talking about a range of numbers. For $|x| < 3$, we're looking for all the numbers whose distance from zero is less than 3. This means numbers between -3 and 3 (not including -3 and 3). So, we can write this as a compound inequality: $-3 < x < 3$. On the other hand, for $|x| > 2$, we're searching for numbers whose distance from zero is greater than 2. This includes all numbers less than -2 or all numbers greater than 2. So, the compound inequality here would be $x < -2$ or $x > 2$. The key takeaway, guys, is that when you're dealing with an absolute value inequality, you'll often end up with two separate inequalities that need to be solved. This is because the expression inside the absolute value could be positive or negative, and both possibilities need to be accounted for to capture all the valid solutions.

Now, let's get back to our specific problem: **solve for $z$ in $1-5|-4 z|

Write a compound inequality like $1 < x < 3$ or like $x < 1$ or $x > 3$. Use integers, proper fractions, or improper fractions in simplest form.** The first step in solving any inequality, absolute value or not, is to isolate the variable term. In this case, our variable term is embedded within the absolute value: $|-4z|$. We want to get this part all by itself on one side of the inequality. So, let's start by subtracting 1 from both sides of the inequality: $1 - 5|-4z|

-1

-5|-4z|

-10$. Now, we need to get rid of that -5 coefficient. We do this by dividing both sides by -5. And here's a super important rule to remember, guys: whenever you multiply or divide an inequality by a negative number, you must flip the direction of the inequality sign. So, $-5|-4z|

-10$ becomes $|-4z|

2$. Excellent! We've successfully isolated the absolute value expression. This is a crucial milestone, and it sets us up perfectly for the next stage of solving.

Breaking Down the Absolute Value

Now that we have $|-4z|

2$, we can finally deal with the absolute value itself. Remember how we talked about the expression inside the absolute value potentially being positive or negative? This is where that comes into play. Since $|-4z|$ must be greater than or equal to 2, it means that the expression $-4z$ must itself be either greater than or equal to 2, or less than or equal to -2. This gives us our two separate inequalities to solve:

1. $-4z

2$ 2. $-4z

-2$

Let's tackle the first one: $-4z

2$. To solve for $z$, we need to divide both sides by -4. And don't forget our golden rule: divide by a negative, flip the inequality sign! So, $-4z

2$ becomes $z

rac{2}{-4}$, which simplifies to $z

-rac{1}{2}$.

Now for the second inequality: $-4z

-2$. Again, divide both sides by -4 and flip that sign: $-4z

-2$ becomes $z

rac{-2}{-4}$, which simplifies to $z

rac{1}{2}$.

So, we have two potential solutions for $z$: $z

-rac{1}{2}$ and $z

rac{1}{2}$. But remember, the original inequality was $|-4z|

2$. This means we're looking for values of $z$ where the absolute value of $-4z$ is greater than or equal to 2. This implies that $-4z$ itself can be quite large in the positive direction or quite large in the negative direction. When we solved $-4z

2$, we found the boundary where $-4z$ starts being at least 2. Values of $z$ smaller than -rac{1}{2} would make $-4z$ more positive than 2, satisfying the condition. For example, if $z = -1$, then $-4z = 4$, and $|4| = 4$, which is indeed $

2$. Conversely, when we solved $-4z

-2$, we found the boundary where $-4z$ is at most -2. Values of $z$ larger than rac{1}{2} would make $-4z$ more negative than -2, also satisfying the condition. For instance, if $z = 1$, then $-4z = -4$, and $|-4| = 4$, which is also $

2$. This understanding is crucial for constructing the correct compound inequality.

Constructing the Compound Inequality

We've done the heavy lifting, guys! We've isolated the absolute value and broken it down into two separate inequalities. Now, it's time to put it all together and write our final answer as a compound inequality. We found that $z

-rac{1}{2}$ and $z

rac{1}{2}$.

Because our original inequality was $|-4z|

2$, which means the expression $-4z$ must be greater than or equal to 2 OR less than or equal to -2, our solution for $z$ reflects this