Problem 42 - College Algebra

Question

Graph both straight lines (left-hand side being \(y_{1}\) and right-hand side being \(y_{2}\) ) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the \(y\) -values of the lines. $$ 4 x+1<\frac{1}{2} x+3 $$

Step-by-Step Solution

Answer

The point of intersection is \( \left( \frac{4}{7}, \frac{23}{7} \right) \), and the inequality is true for \( x < \frac{4}{7} \).

Step 1: Identify Equations

First, we identify the equations for the two lines. Let \( y_1 = 4x + 1 \) and \( y_2 = \frac{1}{2}x + 3 \). We will graph these lines to find their intersection point.

Step 2: Graph the Lines

Graph the lines \( y_1 = 4x + 1 \) and \( y_2 = \frac{1}{2}x + 3 \). Line \( y_1 \) has a y-intercept of 1 and a slope of 4, so it rises quickly. Line \( y_2 \) has a y-intercept of 3 and a slope of \( \frac{1}{2} \), so it rises more slowly.

Step 3: Find the Point of Intersection

To find the point of intersection, set \( y_1 = y_2 \), resulting in the equation \( 4x + 1 = \frac{1}{2}x + 3 \). Solve for \( x \) by isolating \( x \) on one side: \[ 4x - \frac{1}{2}x = 3 - 1 \] which simplifies to \[ \frac{7}{2}x = 2 \]. Solving gives \( x = \frac{4}{7} \). Substitute \( x = \frac{4}{7} \) back into either equation to find \( y \). Using \( y_1 \): \( y = 4(\frac{4}{7}) + 1 = \frac{16}{7} + \frac{7}{7} = \frac{23}{7} \). The point of intersection is \( \left( \frac{4}{7}, \frac{23}{7} \right) \).

Step 4: Analyze the Inequality

We need to determine where \( 4x + 1 < \frac{1}{2}x + 3 \) by comparing \( y_1 \) and \( y_2 \). Rearrange the inequality: \( 4x + 1 < \frac{1}{2}x + 3 \) becomes \( 4x - \frac{1}{2}x < 2 \), which simplifies to \( \frac{7}{2}x < 2 \). Solving this gives \( x < \frac{4}{7} \).

Step 5: Interpret the Solution

The inequality \( 4x + 1 < \frac{1}{2}x + 3 \) is satisfied for \( x < \frac{4}{7} \). On the graph, to the left of the intersection point \( \left( \frac{4}{7}, \frac{23}{7} \right) \), line \( y_1 \) is below line \( y_2 \).

Key Concepts

Graphing Lines

Graphing lines involves plotting a straight line on a coordinate plane, defined by an equation of the form \( y = mx + b \). Here, \( m \) is the slope, which shows how steep the line is, and \( b \) is the y-intercept, the point where the line crosses the y-axis. Understanding how to graph lines is crucial because it helps visualize the relationship between two variables. By graphing the lines \( y_1 = 4x + 1 \) and \( y_2 = \frac{1}{2}x + 3 \), we can see how they interact with each other.
When graphing each line, start by marking the y-intercept on the y-axis. For \( y_1 \), start at \( (0, 1) \), and for \( y_2 \), start at \( (0, 3) \).
Next, use the slope to determine additional points. For \( y_1 \), a slope of 4 means for every 1 unit moved on the x-axis, go up 4 units on the y-axis. For \( y_2 \), with a slope of \( \frac{1}{2} \), go up 0.5 units for every 1 unit moved on the x-axis.
Draw straight lines through these points to complete the graph. This visual representation aids in identifying where the lines intersect and understanding the inequality solution.

Point of Intersection

The point of intersection between two lines is where they cross each other on a graph. To find this point, set the equations for the two lines equal to each other because their y-values are the same at this intersection.
For the equations \( y_1 = 4x + 1 \) and \( y_2 = \frac{1}{2}x + 3 \), equate them to find the intersection:

• Set \( 4x + 1 = \frac{1}{2}x + 3 \)
• Rearrange to \( 4x - \frac{1}{2}x = 3 - 1 \)
• Simplify to \( \frac{7}{2}x = 2 \)
• Solve for \( x \), giving \( x = \frac{4}{7} \)

Substitute \( x = \frac{4}{7} \) back into either line equation to find the y-coordinate. Using \( y_1 \), where \( y = 4(\frac{4}{7}) + 1 = \frac{23}{7} \), thus, the point of intersection is \( \left( \frac{4}{7}, \frac{23}{7} \right) \).
This point is crucial in determining the solution for the inequality, as it defines the boundary where one line is above or below the other.

Slope and Y-intercept

The slope of a line tells us the direction and steepness of the line, calculated as "rise over run"—the change in y divided by the change in x. The slope (m) highlights how much y changes for each increase of 1 in x.
The y-intercept is the starting point of the line on the y-axis, where x equals zero. It’s the "b" in the line equation \( y = mx + b \).
For \( y_1 = 4x + 1 \):

• Slope (m) is 4, indicating a steep rise.
• Y-intercept (b) is 1, showing it crosses the y-axis at \((0, 1)\).

For \( y_2 = \frac{1}{2}x + 3 \):

• Slope (m) is \( \frac{1}{2} \), indicating a gentle rise.
• Y-intercept (b) is 3, showing it crosses the y-axis at \((0, 3)\).

Understanding the slope and y-intercept helps plot lines accurately and reveals how changes in x affect y. The steeper the slope, the faster the increase or decrease in y for a given change in x.