Problem 40 - College Algebra

Question

For the following exercises, graph both straight lines (left-hand side being \(y_{1}\) and right-hand side being \(y_{2}\) ) on the same axes. Find the poin of intersection and solve the inequality by observing where it is true comparing the \(y\) -values of the lines. $$ x+1>x+4 $$

Step-by-Step Solution

Answer

No solution, as the inequality is never true.

Step 1: Identify the Equations

The given inequality is \( x + 1 > x + 4 \). We interpret \( y_1 = x + 1 \) and \( y_2 = x + 4 \) as the equations of two lines.

Step 2: Analyze the Equations

The equations \( y_1 = x + 1 \) and \( y_2 = x + 4 \) both have the same slope of 1 but different intercepts. \( y_1 \) has a y-intercept of 1, and \( y_2 \) has a y-intercept of 4.

Step 3: Determine the Relationship

Upon comparing the intercepts and considering the lines are parallel, \( y_1 = x + 1 \) is always less than \( y_2 = x + 4 \). Thus, \( x + 1 \) can never be greater than \( x + 4 \).

Step 4: Solve the Inequality

Since \( x + 1 \leq x + 4 \) for all real \( x \) and the inequality asks for \( x + 1 > x + 4 \), the inequality \( x + 1 > x + 4 \) has no solution.

Key Concepts

Linear Equations

Linear equations are expressions that form a straight line when graphed on a coordinate plane. They typically take the form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis. In essence, a linear equation shows a constant rate of change between the variables.
Understanding linear equations is crucial because they serve as a foundation for many mathematical concepts, including graphing, system solving, and inequality analysis.
In the given exercise, the equations \(y_1 = x + 1\) and \(y_2 = x + 4\) are linear equations, with each having a slope of 1, signifying that the lines are parallel.

Graphing Lines

Graphing lines involves plotting the equation of a line on a coordinate plane to visually understand its behavior and relationship to other lines or points.
To graph a line, start by identifying the slope and y-intercept from its equation. Plot the y-intercept on the y-axis. Then, using the slope, determine the direction and steepness of the line by moving according to the slope's rise over run. Connect the dots to form a line.
For the exercise at hand, plotting \(y_1 = x + 1\) and \(y_2 = x + 4\) makes it clear that both lines never intersect due to their identical slope, helping directly in analyzing the inequality.

Parallel Lines

Parallel lines are lines in a plane that never meet; they have the same slope but different y-intercepts. This means that no matter how far these lines extend, they remain equidistant from each other all through.
In mathematical terms, if two lines are represented by \(y = m_1x + b_1\) and \(y = m_2x + b_2\), they are parallel if \(m_1 = m_2\).
In the example given, \(y_1 = x + 1\) and \(y_2 = x + 4\) are parallel with both having a slope of 1. Since their intercepts differ, \(y_1\) is consistently below \(y_2\) across the entire graph.

Inequality Analysis

Inequality analysis involves comparing two expressions to find when one is greater, less, or equal to the other. In this context, it’s often represented as a statement involving the symbols of inequality: \(>\), \(<\), \(\geq\), or \(\leq\). Analyzing inequalities often involves graphing equations to visually identify regions of interest.
For the inequality \(x + 1 > x + 4\), use the graphical interpretation of \(y_1\) and \(y_2\). Here, by observing that \(y_2\) is always above \(y_1\) due to their parallel nature, we see clearly that \(x + 1\) can never be greater than \(x + 4\). Hence, this inequality has no solution. Through graph analysis, one can determine whether regions or solutions exist.