Graphing The Inequality Y Less Than 2|x-1|-2 And Identifying Solution Points

Jun 16, 2025 by ADMIN 77 views

Graphing Inequalities with Absolute Values: A Step-by-Step Guide

In mathematics, graphing inequalities is a fundamental skill, especially when dealing with absolute values. This article will guide you through the process of graphing the inequality $y < 2|x-1|-2$ , and we'll also determine which point from a given set is part of the solution. Let's dive in!

Understanding Absolute Value Inequalities

Before we graph, it's crucial to understand absolute value inequalities. The absolute value of a number, denoted by $|x|$ , represents its distance from zero on the number line. This means $|x|$ is always non-negative. When we have an inequality involving absolute values, it often translates into two separate inequalities. The key to solving absolute value inequalities lies in recognizing that the expression inside the absolute value can be either positive or negative, and we need to consider both possibilities.

In our case, we have the inequality $y < 2|x-1|-2$ . The absolute value expression is $|x-1|$ . This means we need to consider two cases:

When $x-1$ is non-negative (i.e., $x-1 extgreater= 0$ or $x extgreater= 1$ ), the absolute value $|x-1|$ simplifies to $x-1$ .
When $x-1$ is negative (i.e., $x-1 < 0$ or $x < 1$ ), the absolute value $|x-1|$ simplifies to $-(x-1)$ , which is equivalent to $1-x$ .

By breaking down the absolute value, we can rewrite our inequality as two separate inequalities, each representing a different region of the graph. This is a critical step in accurately graphing absolute value inequalities.

Step-by-Step Graphing Process

1. Break Down the Absolute Value

As mentioned earlier, the first step is to break down the absolute value into two cases. This allows us to deal with simpler linear inequalities. For our inequality $y < 2|x-1|-2$ , we have:

Case 1: $x extgreater= 1$ . In this case, $|x-1| = x-1$ , so the inequality becomes:

$y < 2(x-1) - 2$

$y < 2x - 2 - 2$

$y < 2x - 4$
Case 2: $x < 1$ . In this case, $|x-1| = 1-x$ , so the inequality becomes:

$y < 2(1-x) - 2$

$y < 2 - 2x - 2$

$y < -2x$

Now we have two linear inequalities: $y < 2x - 4$ for $x extgreater= 1$ and $y < -2x$ for $x < 1$ . These are much easier to graph than the original inequality with the absolute value.

2. Graph the Boundary Lines

Next, we graph the boundary lines for each inequality. The boundary lines are the lines obtained by replacing the inequality sign with an equals sign. For our two inequalities, the boundary lines are:

$y = 2x - 4$
$y = -2x

To graph these lines, we can find two points on each line. For $y = 2x - 4$ , we can use the points (2, 0) and (3, 2). For $y = -2x$ , we can use the points (0, 0) and (1, -2). Plot these points on the coordinate plane and draw the lines.

Since our original inequality uses a "less than" sign ( $<$ ), the boundary lines should be dashed lines. This indicates that the points on the lines are not part of the solution. If the inequality had included an "equals" sign ($ extless=$), we would draw solid lines to indicate that the points on the lines are part of the solution. Dashed lines are crucial for representing strict inequalities, while solid lines denote inclusive inequalities.

3. Determine the Shaded Region

The next step is to determine the shaded region for each inequality. This involves choosing a test point that is not on the boundary line and plugging its coordinates into the inequality. If the inequality is true for the test point, then we shade the region containing the test point. If the inequality is false, we shade the other region.

For $y < 2x - 4$ and $x extgreater= 1$ : Let's choose the test point (2, -1). Plugging these coordinates into the inequality, we get:

$-1 < 2(2) - 4$

$-1 < 0$

This is true, so we shade the region below the line $y = 2x - 4$ for $x extgreater= 1$ .
For $y < -2x$ and $x < 1$ : Let's choose the test point (0, -1). Plugging these coordinates into the inequality, we get:

$-1 < -2(0)$

$-1 < 0$

This is true, so we shade the region below the line $y = -2x$ for $x < 1$ .

The shaded regions together represent the solution to the inequality $y < 2|x-1|-2$ . The graph will resemble a "V" shape, typical of absolute value functions, with the shaded region being below the V.

4. Combine the Graphs

Finally, we combine the graphs of the two inequalities. The solution to the original absolute value inequality is the intersection of the shaded regions from both cases. The resulting graph will show the region where $y$ is less than $2|x-1|-2$ . The graph will have a vertex at the point (1, -2), which is where the two lines $y = 2x - 4$ and $y = -2x$ meet if they were extended. The shaded region will be below this V-shaped graph.

Identifying Solution Points

Now that we have the graph, we can identify which points are part of the solution. We are given four points:

A. (-1, -1)
B. (1, 1)
C. (1, 2)
D. (2, 3)

To determine if a point is part of the solution, we plug its coordinates into the original inequality $y < 2|x-1|-2$ and see if the inequality holds true.

A. (-1, -1):

$-1 < 2|-1-1| - 2$

$-1 < 2|-2| - 2$

$-1 < 2(2) - 2$

$-1 < 4 - 2$

$-1 < 2$ (True)

So, (-1, -1) is part of the solution.
B. (1, 1):

$1 < 2|1-1| - 2$

$1 < 2|0| - 2$

$1 < 0 - 2$

$1 < -2$ (False)

So, (1, 1) is not part of the solution.
C. (1, 2):

$2 < 2|1-1| - 2$

$2 < 2|0| - 2$

$2 < 0 - 2$

$2 < -2$ (False)

So, (1, 2) is not part of the solution.
D. (2, 3):

$3 < 2|2-1| - 2$

$3 < 2|1| - 2$

$3 < 2(1) - 2$

$3 < 2 - 2$

$3 < 0$ (False)

So, (2, 3) is not part of the solution.

Therefore, the only point that is part of the solution is A. (-1, -1).

Key Takeaways

Absolute value inequalities require breaking down the absolute value into two cases.
Graph the boundary lines as dashed or solid lines, depending on the inequality sign.
Use test points to determine the shaded region.
The solution to an absolute value inequality is often a V-shaped region.
To check if a point is a solution, substitute its coordinates into the original inequality.

Mastering these steps will enable you to confidently graph and solve a wide range of inequalities, including those involving absolute values. Remember, practice makes perfect, so work through various examples to solidify your understanding.

By following this guide, you can easily graph the inequality $y < 2|x-1|-2$ and determine which points are part of the solution set. This skill is invaluable in various mathematical contexts, from algebra to calculus.