Problem 35 - Algebra A Combined Function

Question

Graph each inequality in two variables. $$ y<2 x-1 $$

Step-by-Step Solution

Answer

Graph a dashed line for \( y = 2x - 1 \) and shade below it.

Step 1: Understand the Inequality

The inequality given is \( y < 2x - 1 \). This is a linear inequality in two variables, \( x \) and \( y \). It represents a region below the line \( y = 2x - 1 \).

Step 2: Graph the Boundary Line

First, graph the boundary line of the inequality, \( y = 2x - 1 \). This line is not part of the solution, so draw it as a dashed line. To find the line, calculate two points, for example:1. When \( x = 0 \), \( y = 2(0) - 1 = -1 \), giving the point (0, -1).2. When \( x = 1 \), \( y = 2(1) - 1 = 1 \), giving the point (1, 1).Draw a dashed line through these points.

Step 3: Shade the Solution Region

The inequality \( y < 2x - 1 \) represents the region below the boundary line. Choose a test point not on the line, such as \( (0,0) \). Substitute into the inequality:\( 0 < 2(0) - 1 \rightarrow 0 < -1 \).This is false, so the test point \( (0,0) \) is not in the solution region. Thus, the region to shade is the one on the opposite side of the line from \( (0,0) \). Shade the region below the dashed line.

Key Concepts

Linear Inequality

A linear inequality is much like a linear equation, but it contains an inequality sign, such as

• less than (\(<\)
• greater than (\(>\)
• less than or equal to (\(\leq\)
• greater than or equal to (\(\geq\)

For example, the inequality given in our problem is \(y < 2x - 1\), which states that the value of \(y\) is less than the value generated by \(2x - 1\) for any given \(x\).Linear inequalities can be visualized as regions on a graph, as opposed to lines. This is due to the inequality sign indicating multiple possible solutions. These solutions are often found to be in a shaded area on the graph that fulfills the conditions of the inequality.

Boundary Line

The boundary line plays a crucial role when graphing linear inequalities. It is formulated by treating the inequality as an equation. For \(y < 2x - 1\), the boundary line is \(y = 2x - 1\).To draw this line, find at least two points:

• Set \(x = 0\) and solve for \(y\): \(y = 2(0) - 1 = -1\), resulting in the point \((0, -1)\).
• Set \(x = 1\) and solve: \(y = 2(1) - 1 = 1\), resulting in the point \((1, 1)\).

The line drawn through these points is dashed because the inequality is strictly less than (\(<\)), not "less than or equal to." This visually communicates that points on the line are not part of the solution.

Solution Region

The solution region is the area where the inequality is satisfied. In this context,\(y < 2x - 1\) means the solution region is everything below the boundary line.To determine which side of the boundary line should be shaded:1. Choose a test point not on the line, commonly \((0, 0)\).2. Substitute into the inequality: - \(0 < 2(0) - 1\) results in\(0 < -1\), which is false.Since the test point is not part of the solution, you must shade the opposite side to where the test point is located. In this case, shade below the boundary line.

Test Point Method

The test point method is an essential technique to verify which section of the graph fulfills the inequality. Use this simple process to correctly shade the graph area that represents the solution.Here's how it works:- Select a reference or test point outside of the suspected solution region. Choosing points like \((0, 0)\) is common as calculations are easier.- Substitute the point into the given inequality.- Analyze the result: - If true, the test point lies within the solution area. Shade the region on the same side as the test point. - If false, shade the opposite side of the boundary line.This method guarantees that the graph displays the correct solution region, visualizing all possible solutions of the inequality.