Problem 39 - 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. $$ x-2>2 x+1 $$

Step-by-Step Solution

Answer

The solution is \( x < -3 \), where \( y_1 \) is greater than \( y_2 \).

Step 1: Identify the Equations

First, identify the equations representing the left-hand side and right-hand side of the inequality. For the left-hand side, we have the line equation: \( y_1 = x - 2 \). For the right-hand side, the line equation is: \( y_2 = 2x + 1 \).

Step 2: Rearrange to Equal Lines

To find where the lines intersect, we set the equations equal: \( x - 2 = 2x + 1 \). This allows us to find the point where both lines meet, which will help us determine where one line is greater than the other.

Step 3: Solve for Intersection

From \( x - 2 = 2x + 1 \), rearrange to get \( x - 2x = 1 + 2 \). Simplify to find \( -x = 3 \), and thus \( x = -3 \). Substitute \( x = -3 \) into one of the line equations (e.g., \( y_1 = x - 2 \)) to get \( y = -5 \). So the intersection point is \((-3, -5)\).

Step 4: Graph the Lines

Plot both lines on the graph. \( y_1 = x - 2 \) has a y-intercept at \( (0, -2) \) and a slope of 1. \( y_2 = 2x + 1 \) has a y-intercept at \( (0, 1) \) and a slope of 2. Also, mark the intersection point \((-3, -5)\).

Step 5: Observe Inequality

The inequality \( x - 2 > 2x + 1 \) implies \( y_1 > y_2 \). On the graph, this condition is precisely where the line for \( y_1 \) (i.e., \( x - 2 \)) is above the line for \( y_2 \) (i.e., \( 2x + 1 \). Since the lines intersect at \( x = -3 \), \( y_1 \) is greater than \( y_2 \) for \( x < -3 \).

Key Concepts

Graphing Lines

Graphing lines is an essential skill in understanding and visualizing linear equations. When we graph a line, we transform an algebraic equation into a visual representation. This helps us identify relationships and interactions between different lines on a coordinate plane.

To graph the line represented by the equation, you need to identify two important features:

• The slope (often denoted as \(m\)) which tells us how steep the line is.
• The y-intercept, which is the point where the line crosses the y-axis.

With these elements, we can start plotting. Let's look into the equations:\( y_1 = x - 2 \) has a slope of 1 and a y-intercept at \((0, -2)\). Meanwhile, \( y_2 = 2x + 1 \) boasts a slope of 2 and crosses the y-axis at \((0, 1)\).

When graphing these on the same axis, plot the points based on their y-intercepts and use the slopes to determine how steeply they ascend or descend across the grid. The line \( y_1 \) starts from -2 and rises gently with a slope of 1. Conversely, \( y_2 \) starts from 1 and climbs sharply with a steeper slope of 2.

Points of Intersection

Points of intersection occur where the lines, represented by their respective equations on a graph, meet or cross each other. This concept is crucial in solving systems of equations, including inequalities.

In order to find a point of intersection algebraically, we treat the equations as if they are equal to one another and solve for \(x\), as demonstrated in the solution steps:

Given the equations \( y_1 = x - 2 \) and \( y_2 = 2x + 1 \), they intersect where \( x - 2 = 2x + 1 \). Solving this gives us the intersection point \((x, y) = (-3, -5)\).

Why is finding this point important? It helps us analyze where one line overtakes the other, invaluable when interpreting inequalities. In this exercise, \( y_1 \) is greater than \( y_2 \) for values of \(x\) less than -3. Therefore, identifying the intersection enables us to effectively solve the inequality.

Linear Equations

Linear equations form the foundation of many algebraic concepts, including inequalities and their graphical representations. A linear equation takes the form \( y = mx + b \), where \(m\) is the slope, and \(b\) is the y-intercept.

These equations create straight lines that can be plotted on a graph to reveal relationships between different variables. In our exercise, understanding the component parts of each equation allows us to graph and interpret their interactions effectively.

For \( y_1 = x - 2 \), the slope \(m = 1\) indicates that for every step in \(x\), \(y\) increases by 1. The line crosses the y-axis at \(-2\). Meanwhile, \( y_2 = 2x + 1 \) has a slope \(m = 2\), meaning \(y\) increases by 2 for every step in \(x\), and the line crosses the y-axis at 1.

By plotting these equations, we can identify intersections and solve inequalities like \( x - 2 > 2x + 1 \) by observing where \( y_1 \) surpasses \( y_2 \) on the graph.