Problem 48 - Algebra 2

Question

Graph each inequality. $$ y>|5 x| $$

Step-by-Step Solution

Answer

Shade the region above the V-shaped graph of \(y=|5x|\) using dashed lines.

Step 1: Understand the Absolute Value Function

The inequality involves an absolute value, \(|5x|\), which means we need to understand its graph. The graph of \(y = |5x|\) is a V-shaped curve with the vertex at the origin (0,0), and it opens upwards. This is because absolute values result in non-negative outputs only.

Step 2: Identify the Region for the Inequality

The inequality is \(y > |5x|\), which means we are interested in the region above the graph of \(y = |5x|\). This region will include all points where the y-coordinate is greater than the V-shaped curve.

Step 3: Plot the Parent Graph

Begin by drawing the graph of \(y = |5x|\). This involves plotting the line \(y = 5x\) for \(x \geq 0\) and \(y = -5x\) for \(x < 0\). This creates a V-shape with the vertex at the origin.

Step 4: Shade the Desired Region

Since we want \(y\) to be greater than \(|5x|\), shade the area above the V-shaped graph drawn in Step 3. This area represents all the points where the y-coordinate is greater than the absolute value.

Step 5: Consider the Type of Line to Use

When plotting inequalities, use a dashed line when the inequality is strict (\(>\) or \(<\)). Therefore, use a dashed line for \(y = |5x|\) to indicate the boundary isn't included in the solution set.

Key Concepts

Absolute Value Function

At the core of this exercise is the absolute value function, which plays a crucial role in shaping the graph of an inequality. Here, we focus on the function \( y = |5x| \). Absolute value functions are known for their distinctive V-shaped graphs, reflecting the non-negative characteristic of absolute values.

• The simplest form of an absolute value graph is \( y = |x| \), which has its vertex at the origin (0,0).
• The absolute value function \( y = |5x| \) stretches vertically compared to \( y = |x| \) because of the factor of 5.
• It remains symmetrical about the y-axis due to the absolute value.

Overall, it's crucial to anticipate the V-shape and understand that it results from the absolute value taking in any negative input and returning its positive equivalent.

Vertex of the Graph

The vertex of the graph is a pivotal element for understanding the shape and structure of the V-shaped graph produced by an absolute value function. For the inequality \( y > |5x| \), the vertex is situated at the origin (0,0).

• The vertex is the point where the graph changes direction, forming the base of the V-shape.
• In absolute value functions, the vertex represents the minimum point for expressions such as \( y = |5x| \), as y cannot be negative.

Knowing the location and significance of the vertex will help in both sketching the graph and understanding the regions of interest when solving inequalities.

Shading in Graphs

Shading is an essential technique when graphing inequalities, as it visually distinguishes the solution set. For the inequality \( y > |5x| \), shading the appropriate area is essential.

• Since the inequality is \( y > |5x| \), the region above the V-shaped graph represents all the solutions.
• Shading begins directly above the V, indicating all points where the y-coordinate is greater than the value on the line.

Shading not only makes it easier to see the solution set but also helps differentiate between solutions involving strict inequalities versus those with equality.

Dashed Lines in Graphs

Using dashed lines in graphs is a crucial detail when representing strict inequalities. In this exercise, where we graph \( y > |5x| \), we employ a dashed line instead of a solid one.

• Dashed lines indicate that points on the line \( y = |5x| \) are not part of the solution set.
• The inequality signs \( > \) or \( < \) direct the use of dashed lines, showing that the boundary itself is excluded from the solutions.

In contrast, a solid line would imply that points on the boundary are included. Therefore, knowing when to use dashed lines is key to accurately graphing inequalities.