Graphing Inequalities: Amber's Step-by-Step Solution

Aug 8, 2025 by Luna Greco 53 views

Solving Inequalities Graphically: A Guide to Amber's Approach

Hey guys! Today, we're diving into the world of inequalities and how to solve them graphically. Specifically, we're going to tackle the problem Amber is facing: solving the inequality $|x+6|-12<13$ by graphing. This might sound a bit intimidating at first, but trust me, it's super manageable once we break it down step by step. We'll explore the fundamental concepts, walk through the solution, and highlight why Amber needs to graph specific equations to conquer this inequality. So, buckle up and let's get started!

Understanding the Inequality: $|x+6|-12<13$

Before we even think about graphing, let's make sure we understand what the inequality $|x+6|-12<13$ is telling us. The core concept here is the absolute value, denoted by the vertical bars $|...|$ . Remember, the absolute value of a number is its distance from zero, regardless of direction. So, $|x+6|$ represents the distance of $(x+6)$ from zero. This means $|x+6|$ will always be non-negative.

The inequality states that this distance, $|x+6|$ , after having 12 subtracted from it, must be less than 13. To solve this, our mission is to find all the values of 'x' that make this statement true. A graphical approach offers a visual way to achieve this.

To solve this absolute value inequality, we need to isolate the absolute value expression. We can do this by adding 12 to both sides of the inequality:

$|x+6|-12<13$

Adding 12 to both sides, we get:

$|x+6| < 25$

Now, this inequality is in a much more manageable form. It tells us that the distance of $(x+6)$ from zero must be less than 25. This is the key to figuring out which equations Amber should graph.

The Graphical Approach: Visualizing the Solution

The beauty of graphing inequalities lies in its visual representation. Instead of just crunching numbers, we can see the solution set. To solve the inequality $|x+6| < 25$ graphically, we'll treat each side of the inequality as a separate function:

Left-hand side: $y_1 = |x+6|$
Right-hand side: $y_2 = 25$

By graphing these two equations, we can visually identify the regions where the graph of $y_1 = |x+6|$ is below the graph of $y_2 = 25$ . Why below? Because our inequality states that $|x+6|$ (which is $y_1$ ) must be less than 25 (which is $y_2$ ).

The graph of $y_1 = |x+6|$ is a V-shaped graph. The basic absolute value function, $y=|x|$ , has its vertex (the pointy bottom part) at the origin (0,0). The “+6” inside the absolute value shifts the graph 6 units to the left. So, the vertex of our graph will be at (-6,0).

The graph of $y_2 = 25$ is a horizontal line at y=25. It's a straight line that runs parallel to the x-axis, intersecting the y-axis at 25.

The solution to the inequality will be the x-values where the V-shaped graph of $y_1 = |x+6|$ is below the horizontal line $y_2 = 25$ . To find these x-values, we need to identify the points where the two graphs intersect. These intersection points mark the boundaries of our solution set.

Why These Equations? The Core of the Solution

So, why exactly do we need to graph $y_1 = |x+6|$ and $y_2 = 25$ ? Let's break it down:

$y_1 = |x+6|$ : This equation represents the left-hand side of our simplified inequality, $|x+6| < 25$ . It captures the absolute value expression, which is crucial for understanding the distance concept. Graphing this function allows us to visualize how the distance of $(x+6)$ from zero changes as 'x' varies.
$y_2 = 25$ : This equation represents the right-hand side of our simplified inequality. It provides the benchmark value. We're interested in the regions where the absolute value expression is less than this value. Graphing this as a horizontal line gives us a clear visual comparison.

By graphing these two equations together, we create a visual landscape where we can directly compare the values of $|x+6|$ and 25. The regions where the V-shaped graph is below the horizontal line directly correspond to the x-values that satisfy the inequality.

Amber's Choice: Identifying the Correct Equations

Now that we understand the graphical approach, let's revisit the options Amber has:

A. $y_1=|x+6|, y_2=25$ B. $y_1=x+6, y_2=25$ C. $y_1=|x+6|, y_2=13$ D. $y_1=x+6, y_2=13$

Based on our discussion, we know Amber needs to graph the equations that represent the absolute value expression and the constant value from the simplified inequality, $|x+6| < 25$ . Therefore, the correct equations are:

$y_1 = |x+6|$
$y_2 = 25$

This corresponds to option A. Options B and D are incorrect because they graph $y_1 = x+6$ instead of the absolute value expression $y_1 = |x+6|$ . Option C is incorrect because it graphs $y_2 = 13$ , which is the constant from the original inequality before we isolated the absolute value.

Solving the Inequality: Beyond the Equations

Choosing the right equations is just the first step. Once Amber graphs $y_1 = |x+6|$ and $y_2 = 25$ , she needs to do the following to completely solve the inequality:

Identify the Intersection Points: Find the x-coordinates where the two graphs intersect. These points are crucial because they define the boundaries of the solution set.
Determine the Region: Look for the region on the graph where the graph of $y_1 = |x+6|$ is below the graph of $y_2 = 25$ . This region represents the x-values that satisfy the inequality.
Express the Solution: Write the solution as an inequality or in interval notation. The solution will be the set of all x-values within the region identified in step 2.

To actually find the intersection points and the solution interval, we can solve the absolute value inequality algebraically. Remember that $|x+6| < 25$ means that the distance of $(x+6)$ from zero is less than 25. This translates to two separate inequalities:

$-25 < x+6 < 25$

To solve this compound inequality, we subtract 6 from all three parts:

$-25 - 6 < x+6 - 6 < 25 - 6$
$-31 < x < 19$

Therefore, the solution to the inequality $|x+6| < 25$ is $-31 < x < 19$ . In interval notation, this is written as $(-31, 19)$ . This means that all x-values between -31 and 19 (excluding -31 and 19) will satisfy the original inequality.

Graphically, this means that the V-shaped graph of $y_1 = |x+6|$ will be below the horizontal line $y_2 = 25$ for all x-values between -31 and 19. The intersection points of the two graphs will be at x = -31 and x = 19.

Key Takeaways for Graphing Inequalities

Let's recap the key concepts we've learned today:

Isolate the Absolute Value: Before graphing, make sure to isolate the absolute value expression in the inequality.
Treat Each Side as a Function: Consider each side of the inequality as a separate function (y1 and y2).
Graph the Functions: Graph both functions on the same coordinate plane.
Identify the Region: Determine the region where the graph of y1 satisfies the inequality (e.g., below, above).
Find the Intersection Points: The x-coordinates of the intersection points define the boundaries of the solution.
Express the Solution: Write the solution as an inequality or in interval notation.

By following these steps, you can confidently tackle inequalities using a graphical approach. Remember, the visual representation provides a powerful way to understand and solve these problems.

Conclusion: Amber's Success and Your Journey

So, there you have it! We've helped Amber (and hopefully you!) understand how to solve the inequality $|x+6|-12<13$ by graphing. The key is to isolate the absolute value, treat each side as a function, graph the functions, and identify the relevant region. Remember, option A, graphing $y_1=|x+6|$ and $y_2=25$ , is the correct approach.

Solving inequalities graphically is a valuable skill in mathematics. It not only provides a visual understanding of the solution but also reinforces the concepts of functions and inequalities. Keep practicing, and you'll become a pro at solving inequalities graphically in no time! Keep exploring, keep learning, and most importantly, have fun with math! You guys got this!