Problem 28 - Algebra for College Students

Question

\(\operatorname{Graph}|y|>1\).

Step-by-Step Solution

Answer

Graph two shaded regions: one above y=1 and one below y=-1.

Step 1: Understand the Inequality

The inequality \(|y| > 1\) means that the absolute value of \(y\) is greater than 1. This implies that \(y\) must either be greater than 1 or less than -1.

Step 2: Break Down the Absolute Value Inequality

To solve \(|y| > 1\), we write it as two separate inequalities: \(y > 1\) or \(y < -1\). These conditions show the regions where the absolute value inequality holds true.

Step 3: Graph y > 1 on the Coordinate Plane

On a graph, \(y > 1\) represents a region above the line \(y = 1\). This region does not include the line itself, so it should be represented as a dashed line for \(y=1\).

Step 4: Graph y < -1 on the Coordinate Plane

Similarly, \(y < -1\) represents a region below the line \(y = -1\). Like the previous graph, this should also be represented as a dashed line to show that the region does not include the line \(y = -1\).

Step 5: Combine and Shade the Graph

Shade the regions that satisfy either condition \(y > 1\) or \(y < -1\). The graph will have two separate shaded areas: above the dashed line \(y = 1\) and below the dashed line \(y = -1\). The area between these lines (including the lines themselves) remains unshaded because \(|y| \leq 1\) in that interval.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane, often called the Cartesian plane, is a two-dimensional surface where we can graphically represent equations and inequalities. It consists of two perpendicular number lines: the x-axis and the y-axis. These axes intersect at a point called the origin, which has coordinates (0,0). Each point on the plane is identified by an ordered pair of numbers representing its position relative to the axes. This makes the coordinate plane invaluable for visualizing mathematical relationships.

When graphing absolute value inequalities like \(|y| > 1\), the coordinate plane allows us to clearly see all the solutions. By graphing lines or curves and shading certain regions, we can better understand which parts of the plane satisfy the inequality. In this case, \(|y| > 1\) involves graphing regions related to y-values, showcasing areas on the plane that satisfy these conditions.

Solving Inequalities and Identifying Regions of Solutions

When you solve the inequality \(|y| > 1\), you break it down into two simpler inequalities: \(y > 1\) and \(y < -1\). This separation is key to identifying the regions of the plane where the inequality holds true.

Each inequality describes a particular region on the coordinate plane:

• For \(y > 1\), the region involves all points above the line \(y = 1\).
• For \(y < -1\), the region includes all points below the line \(y = -1\).

Both these conditions describe areas on the coordinate plane where the solutions to \(|y| > 1\) reside. By shading the areas above \(y = 1\) and below \(y = -1\), you effectively mark these regions of solutions, leaving the region where \(-1 \leq y \leq 1\) unshaded. This graphically communicates that no solutions exist between \(y = 1\) and \(y = -1\).

Graphical Representation of Inequalities

Graphing inequalities on the coordinate plane is a powerful method to visualize solutions. For the inequality \(|y| > 1\), the graphical representation involves certain steps:

• First, draw dashed lines for \(y = 1\) and \(y = -1\) because the inequalities are strict (they use ">", not "≥"). This indicates the boundaries are not part of the solution.
• Second, identify and shade the regions where the inequality holds. This involves shading above the line \(y = 1\) and below the line \(y = -1\).

The unshaded area, the one between the two horizontal dashed lines, indicates that none of its points satisfy \(|y| > 1\). The visual distinction on the plane provides an intuitive understanding of where solutions lie, helping to grasp the concept of inequality in a clear and concrete way.