Problem 14 - Thinking Mathematically

Question

Graph each linear inequality. \(y \leq 2 x-1\)

Step-by-Step Solution

Answer

The graph would be of the line \(y = 2x - 1\) but because we have an inequality, we must shade an area. The area below the line, including the line itself, would be shaded since the inequality is \(y \leq 2x-1\).

Step 1: Convert the inequality to the form of a linear equation

The given inequality is \(y \leq 2 x-1\), which can be seen as a linear equation \(y = 2x - 1\). This equation represents a straight line in the Cartesian coordinate system.

Step 2: Plot the line

On a graph, plot the line using the equation \(y = 2x - 1\). Here, the gradient of the line is 2 and the y-intercept is -1. Hence the line would cut the y-axis at -1 and has a slope of 2.

Step 3: Identify and Shade the area

The inequality is \(y \leq 2x - 1\). This means y is equal to or less than \(2x - 1\). Therefore, the solution set will be below the line \(y = 2x - 1\). So, shade the area which is below the given line to show that all points in this region satisfy the inequality.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting a region on a graph that represents all the solutions of an inequality like our example, \(y \leq 2x - 1\).
Understanding how to graph these inequalities is crucial in visualizing solutions rather than just calculating them.
To begin graphing an inequality, we first convert it into a linear equation to find its boundary line, as shown in the original solution steps.

• Start with the inequality: turn it into a simple linear equation.
• Graph the boundary line: use known techniques for plotting linear equations.
• Identify the region: decide which side of the line represents the solutions.
• Shade the area: this visually confirms all the points that are solutions to the inequality.

Keep in mind that if the inequality symbol is "\(<\)" or "\(>\)," the boundary line is dashed to indicate the line itself is not part of the solution set. Conversely, for "\(\leq\)" or "\(\geq\)," it's solid, showing it's part of the solutions too.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional framework created by René Descartes. It’s comprised of two axes: the x-axis (horizontal) and the y-axis (vertical). Both axes intersect at the origin, where the coordinates are (0,0). This system is widely used for graphing equations and inequalities.
Each point in this system is represented as an ordered pair (x, y), with 'x' being the horizontal distance from the origin and 'y' being the vertical.
When graphing a line such as \(y = 2x - 1\) within this system:

• The y-intercept is the point where the line crosses the y-axis. For \(y = 2x - 1\), it's at (0, -1).
• Every move along the x-axis causes a proportional move along the y-axis depending on the line's slope.
• This relationship enables us to easily translate algebraic expressions into geometric representations on this grid.

Clearly understanding how to interpret points and graph functions within this coordinate plane is essential for accurately displaying inequalities.

Linear Equations

Linear equations form straight lines when plotted on a graph. They are typically expressed in the form \(y = mx + c\), where 'm' is the slope and 'c' is the y-intercept. These equations are foundational in graphing as they represent everything from simple lines to inequalities.
In order to graph the inequality \(y \leq 2x - 1\), it is essential to first understand the linear equation \(y = 2x - 1\):

• The slope (m = 2): Indicates the steepness and direction of the line. A slope of 2 means that for every unit increase in x, y increases by 2 units.
• The y-intercept (c = -1): The point where the line cuts the y-axis, indicating the value of y when x is zero.

By translating this equation into a line within the Cartesian Coordinate System, we can then explore where the region lies for the original inequality. Establishing the line opens up understanding as to where all its potential solutions could lie.