Problem 7 - Intermediate Algebra

Question

a. To graph the inequality \(y>3 x-1\), we begin by graphing the boundary line \(y=3 x-1 .\) What is the slope \(m\) of the line? What is its \(y\) -intercept? b. To graph the inequality \(2 x+3 y \leq-6,\) we begin by graphing the boundary line \(2 x+3 y=-6 .\) What are its \(x\) - and \(y\) -intercepts?

Step-by-Step Solution

Answer

For part a, slope is 3, y-intercept is -1. For part b, x-intercept is (-3, 0), y-intercept is (0, -2).

Step 1: Identify the Slope and Y-Intercept for Part a

The inequality given is \( y > 3x - 1 \). First, we graph the boundary line, which is \( y = 3x - 1 \). This equation is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 3 \) and \( b = -1 \).

Step 2: State Slope and Y-Intercept for Part a

The slope \( m \) of the line is \( 3 \), and its y-intercept is \( -1 \).

Step 3: Convert to Standard Form for Part b

The inequality given is \( 2x + 3y \leq -6 \). We begin by graphing the boundary line, \( 2x + 3y = -6 \). This equation is in standard form.

Step 4: Find X-Intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 2x + 3y = -6 \): \[ 2x + 3(0) = -6 \] \[ 2x = -6 \] \[ x = -3 \]. The x-intercept is \( (-3, 0) \).

Step 5: Find Y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 2x + 3y = -6 \): \[ 2(0) + 3y = -6 \] \[ 3y = -6 \] \[ y = -2 \]. The y-intercept is \( (0, -2) \).

Step 6: State X- and Y-Intercepts for Part b

The x-intercept of the boundary line \( 2x + 3y = -6 \) is \( (-3, 0) \), and the y-intercept is \( (0, -2) \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to express the equation of a straight line. It is represented as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. This form is particularly useful because it immediately gives you important information about the line.
For example, in the equation \( y = 3x - 1 \), the number 3 represents the slope, which tells you how steep the line is. The slope can be thought of as "rise over run," or how much the line goes up (or down) for a unit step to the right. More simply, it tells you how much \( y \) changes as \( x \) increases by 1.
On the other hand, the y-intercept in this equation is -1. This value indicates where the line crosses the y-axis, specifically at the point \( (0, -1) \). Starting at this point, you can use the slope to find other points on the line.

X-Intercepts

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept, you simply set \( y = 0 \) in the line's equation and solve for \( x \).
For instance, with the line given by the equation \( 2x + 3y = -6 \), setting \( y = 0 \) allows us to find the x-intercept:

• \( 2x + 3(0) = -6 \)
• \( 2x = -6 \)
• \( x = -3 \)

Thus, the x-intercept is at the point \( (-3, 0) \). This concept is crucial as it provides one of the coordinates necessary for sketching the line on a graph, helping to visualize the boundary of an inequality when converting it from standard form.

Y-Intercepts

Much like the x-intercept, the y-intercept is a fundamental point. It is the location where a line crosses the y-axis and this happens when \( x \) is zero. To find the y-intercept, set \( x = 0 \) in the line's equation and solve for \( y \).
For the line given by \( 2x + 3y = -6 \), we can find the y-intercept by:

• Setting \( x = 0 \)
• \( 2(0) + 3y = -6 \)
• \( 3y = -6 \)
• \( y = -2 \)

Thus, the y-intercept is \( (0, -2) \). Knowing the y-intercept is vital because it provides a starting point on the graph from which we can apply the slope to find other points on the line. It effectively anchors the line on one of the axes, making it easier to sketch accurately.