Problem 46 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ y

Step-by-Step Solution

Answer

Shade the area below the dashed line \( y = x \).

Step 1: Understand the inequality

The inequality given is \( y < x \). This means that for any point on the graph, the \( y \)-coordinate must be less than the \( x \)-coordinate.

Step 2: Identify the boundary line

The boundary for the inequality \( y < x \) is the line \( y = x \). This is because \( y = x \) represents the points where \( y \) is exactly equal to \( x \).

Step 3: Draw the boundary line

Plot the line \( y = x \) on a coordinate grid. This line is a diagonal line passing through the origin with a slope of 1, meaning it rises one unit for every unit it moves to the right. Since the inequality is \( y < x \), this line should be dashed to indicate that points on the line are not included in the solution.

Step 4: Determine the solution region

The inequality \( y < x \) indicates that the solution set is all the points below the \( y = x \) line. To confirm this, choose a test point, such as \( (0, -1) \), and check if it satisfies the inequality. Since \( -1 < 0 \), this point is part of the solution, confirming that the region below the line is correct.

Step 5: Shade the solution region

Shade the region below the dashed line \( y = x \) on the graph. This shaded area represents all the points that satisfy the inequality \( y < x \).

Key Concepts

Graphing Linear Inequalities

Graphing a linear inequality, like \( y < x \), is similar to graphing a linear equation, but with an added step of shading a region on the graph. First, you need to think of the inequality as an equation. In this case, you would think of \( y < x \) as \( y = x \). This will help you to determine the boundary line. Once you have this equation, you can sketch the boundary line on a coordinate grid.
A key characteristic of these lines is that they divide the graph into two halves, and one of these halves is the solution to the inequality. After drawing the boundary line, you need to determine which side of the line contains all the solutions that satisfy the inequality.

• Start by plotting the boundary line using the linear equation you derived from the inequality.
• Then, decide if this boundary should be a solid or dashed line based on whether the points on the line are included in the inequality.

Boundary Lines in Inequalities

In the context of linear inequalities, the boundary lines play a crucial role. A boundary line is the line that you graph when you replace the inequality sign with an equal sign. For \( y < x \), the boundary line is \( y = x \).
Determining whether to draw a dashed or solid boundary line is important.

• If the inequality is strict, like \( y < x \) or \( y > x \), the boundary line is dashed. This signifies that the points on the line are not part of the solution.
• If it includes equality, like \( y \leq x \) or \( y \geq x \), the line is solid. This means points on the line do satisfy the inequality.

Boundary lines essentially help define the edge of the solution region, splitting the graph into points that either satisfy or do not satisfy the inequality.

Solution Regions in Inequalities

Solution regions are the areas on a graph that represent all the possible solutions to a given inequality. For the inequality \( y < x \), once the boundary line is drawn, you must determine which side of this line contains all the solutions.
A simple way to find this region is to use a test point. Choose a point not on the boundary line, like the origin \((0, 0)\), and plug it into the inequality. If it holds true, the side with this point is part of the solution region. If not, the opposite half plane is the solution.

• In the example of \( y < x \), after plotting the dashed line \( y = x \), choose a test point, such as \( (0, -1) \), to check which side of the line to shade.
• If \(-1 < 0\), this point satisfies the inequality, confirming that the solution region is below the line.

The solution region is generally shaded on the graph to clearly indicate the set of points that satisfy the inequality.