Problem 63 - Introductory Algebra for College Students

Question

Graph the given inequality in a rectangular coordinate system. $$y \geq x+1$$

Step-by-Step Solution

Answer

The graph of the inequality \( y \geq x + 1 \) is represented by the region above the line \( y = x + 1 \) in a rectangular coordinate system, including the line itself. Any point in this region or on the line satisfies the given inequality.

Step 1: Plot the Line

First, treat the inequality as an equation and sketch the line \( y = x + 1 \). This is a straight line that cuts the vertical axis at \( y = 1 \) and has a slope of 1, meaning that for each unit increase in \( x \), \( y \) also increases by 1 unit. This line includes all the points where \( y \) equals \( x + 1 \).

Step 2: Shade Areas

Since the inequality is 'greater than or equal to', it indicates that we want all 'y-values' greater than 'x + 1'. On the coordinate grid, this represents all the areas above the line \( y = x + 1 \). Therefore, shade all the region above the straight line, which includes the line itself (since 'equal to' is part of the inequality).

Step 3: Identify Solution Region

Now, any point in the shaded region above the line \( y = x + 1 \) is a solution to the inequality and fulfils the condition \( y \geq x + 1 \). Therefore, the shaded region on the graph, including the line \( y = x + 1 \), is the solution to this inequality.

Key Concepts

Rectangular Coordinate System

A rectangular coordinate system is an essential tool in algebra and calculus for graphing equations and inequalities. This system is composed of two perpendicular lines, known as axes. The horizontal line is called the x-axis, and the vertical line is the y-axis. Together, these axes create a grid that extends infinitely in all directions.
Any point on this grid can be defined using an ordered pair

• the first element of the pair represents the x-coordinate (position along the x-axis)
• the second element is the y-coordinate (position along the y-axis)

The point where the x-axis and y-axis intersect is called the origin, denoted as (0,0). It is crucial for understanding how to plot equations because every equation is graphed in relation to the origin.
Using this system is fundamental when graphing inequalities, like in the exercise where you graph the inequality \(y \geq x + 1\). This involves using the axes to interpret how the inequality behaves relative to the entire plane. Understanding the coordinate system lets you plot lines precisely and determine which areas need shading for inequalities.

Linear Equations

Linear equations are a type of equation that graph as straight lines on a coordinate grid. A standard form of a linear equation is \(y = mx + b\), where

• \(m\) represents the slope
• \(b\) represents the y-intercept

The slope indicates how steep the line is, with \(m = 1\) meaning that for each step to the right, you move 1 step up.
In the exercise, the inequality \(y \geq x + 1\) is converted first into the equation \(y = x + 1\). Here, the slope \(m\) is 1, and the y-intercept \(b\) is also 1. This means the line starts at the point (0,1) on the y-axis and rises one unit for every unit it moves horizontally to the right.
Linear equations are straightforward to plot because they rely on simple arithmetic and their graph always forms a line. When expanded to linear inequalities, these equations also dictate an entire region of the coordinate plane as solutions, making it crucial to first understand the line they are based on.

Solution Region

The solution region on a graph indicates all the points that satisfy an inequality. For the exercise involving \(y \geq x + 1\), once the line \(y = x + 1\) is graphically represented, it's essential to determine where the inequality holds true.
This involves shading the area above the line because all the y-values in this region are greater than or equal to \(x + 1\). The line itself is included in the shading because the inequality is 'greater than or equal to', meaning points on the line also satisfy the condition. Here’s how to determine the solution region:

• First, plot the line \(y = x + 1\) on the coordinate grid.
• Then, because the inequality indicates 'greater than', shade everything above the line.
• Since it's also 'equal to', the line counts as part of the solution.

Any point within the shaded area, when plugged into the inequality, will make the statement true. Practicing with these steps helps in visualizing and solving inequalities, as the shaded regions signify all possible solutions.