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

Hey math enthusiasts! Today, we're going to dive into the world of absolute value inequalities and learn how to conquer problems like $|4y - 12| + 4 \leq 20$. Don't worry, it might seem intimidating at first, but trust me, with a little practice and a clear understanding of the steps, you'll be solving these with ease. Absolute value inequalities are a fundamental concept in algebra, and understanding them is crucial for more advanced mathematical topics. So, grab your pencils, and let's get started! This guide will walk you through the process step-by-step, ensuring you grasp the core concepts and can confidently solve these types of problems. We'll break down the problem, explain each step, and provide helpful tips to ensure you're on the right track. Let's begin our mathematical adventure into the realm of absolute value inequalities!

Understanding Absolute Value

Before we jump into solving the inequality, let's quickly recap what absolute value means. Absolute value is the distance of a number from zero on the number line. It's always a non-negative value. Think of it this way: regardless of whether a number is positive or negative, its absolute value is always positive (or zero). We use the symbol $| |$ to denote absolute value. For example, $|3| = 3$ and $|-3| = 3$ because both 3 and -3 are 3 units away from zero. This understanding is crucial because it shapes how we approach solving these inequalities. Recognizing that the absolute value expression can represent two different scenarios is key. This duality is what requires us to consider both positive and negative possibilities when solving.

Step-by-Step Solution: $|4y - 12| + 4 \leq 20$

Alright, now let's get to the meat of the matter: solving the inequality. Here's a breakdown of how to solve $|4y - 12| + 4 \leq 20$ step-by-step. Follow these steps, and you'll be golden!

Isolate the Absolute Value Expression

The first thing we want to do is get the absolute value expression by itself on one side of the inequality. To do this, we need to get rid of that pesky $+ 4$. We do this by subtracting 4 from both sides of the inequality:

$|4y - 12| + 4 - 4 \leq 20 - 4$

This simplifies to:

$|4y - 12| \leq 16$

See? We've isolated the absolute value expression, which is the first major step! This sets the stage for the next crucial steps, allowing us to consider the two possible scenarios that the absolute value represents. Isolating the absolute value is like preparing the ground before planting a seed – it's essential for the process to work correctly.

Create Two Separate Inequalities

Now comes the part where the absolute value definition really kicks in. Because the absolute value of an expression is its distance from zero, we need to consider both the positive and negative possibilities of the expression inside the absolute value bars. Think of it as two separate paths, one where the expression is positive, and one where it's negative. We'll create two inequalities:

1. Positive Case: $4y - 12 \leq 16$ (The expression inside the absolute value is positive or zero.)
2. Negative Case: $4y - 12 \geq -16$ (The expression inside the absolute value is negative. When we remove the absolute value, we reverse the inequality sign and negate the right side.)

This is where the core concept of absolute value shines! Remember that absolute value is always non-negative, which means the value inside the bars could have been either positive or negative before the absolute value was taken.

Solve Each Inequality Separately

Now that we have two separate inequalities, we solve them individually. Let's start with the first one, $4y - 12 \leq 16$:

1. Add 12 to both sides: $4y \leq 28$
2. Divide both sides by 4: $y \leq 7$

Great! Now, let's solve the second inequality, $4y - 12 \geq -16$:

1. Add 12 to both sides: $4y \geq -4$
2. Divide both sides by 4: $y \geq -1$

We have now solved both individual inequalities, which are fundamental components in finding the complete solution set for the initial absolute value inequality. Each inequality provides a boundary for the solution.

Combine the Solutions

We have two solutions: $y \leq 7$ and $y \geq -1$. This means that y must be greater than or equal to -1 and less than or equal to 7. We can write this as a compound inequality:

$-1 \leq y \leq 7$

This tells us that the solution to the original inequality is all the numbers between -1 and 7, including -1 and 7. You can visualize this on a number line, where you'd shade the region between -1 and 7, and place closed circles at -1 and 7 to indicate that these points are included in the solution. This step is vital because it unifies the two results we got from solving the individual inequalities and presents the complete solution set. This provides the solution to the original absolute value inequality.

Graphing the Solution

Visualizing the solution on a number line can be incredibly helpful. Draw a number line and mark the points -1 and 7. Since our inequality includes "equal to", we'll use closed circles (filled-in circles) at -1 and 7. Then, shade the region between -1 and 7. This shaded region represents all the values of y that satisfy the inequality. The graph provides a clear, visual representation of the solution set, making it easier to understand the range of values that make the inequality true.

Verification and Tips

Always check your answer! Pick a value within the solution range (like 0, for example) and plug it back into the original inequality to see if it works. If it does, great! You're on the right track. Also, test values outside the range to ensure they do not satisfy the inequality.

Here are some helpful tips:

• Isolate First: Always isolate the absolute value expression before doing anything else.
• Two Inequalities: Remember to create two separate inequalities (positive and negative cases).
• Flip the Sign: When dealing with the negative case, remember to flip the inequality sign.
• Number Line: Use a number line to visualize the solution.
• Check Your Work: Always plug your answers back into the original equation to verify the solution.

Conclusion

Solving absolute value inequalities can seem a bit tricky at first, but with practice and understanding of the steps, it becomes quite manageable. Remember the key steps: isolate the absolute value, create two inequalities, solve each inequality, and combine the solutions. By following these steps and practicing regularly, you'll become a pro at solving absolute value inequalities in no time! Keep practicing, and don't be afraid to ask for help if you get stuck. You've got this!