Problem 28 - Introductory Algebra for College Students

Question

Graph each inequality. $$y>2 x-1$$

Step-by-Step Solution

Answer

To graph the inequality \(y > 2x-1\), plot the line \(y = 2x-1\) as a dashed line and shade the area above the line.

Step 1: Plotting the Line

First, ignore the inequality symbol and treat it as an equation: \(y = 2x - 1\). This is a linear equation, and can be graphed as a straight line. The slope of the line is 2, and the y-intercept is -1. Therefore, plot the y-intercept at the point (0, -1) on the y-axis and use the slope 2 (which equals 2/1) to find more points. From the y-intercept, move up 2 units (for rise) and run (move right) 1 unit for each additional point.

Step 2: Draw the Line

Since the inequality is \(y > 2x-1\) and not \(y \geq 2x-1\), we know that the line should be dashed, not solid. A dashed line indicates that the points on the line itself are not included in the inequality. Based on the points plotted, draw a dashed straight line.

Step 3: Shade the Correct Area

We shade the area of the graph where the inequality is true. In this case, the inequality is \(y > 2x-1\), so we shade the area above the line, which represents all y-values that are greater than \(2x-1\).

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra. They describe a straight line in the Cartesian coordinate system and are generally expressed in the form \( y = mx + b \). In this formula:

• \( y \) represents the dependent variable or the value we are trying to solve for.
• \( m \) is the slope of the line, depicting how steep the line is.
• \( x \) is the independent variable, a variable that we use to calculate \( y \).
• \( b \) is the y-intercept, which shows where the line crosses the vertical y-axis.

Understanding the parts of the equation helps us predict how changes in one variable affect another. In our example, the linear equation \( y = 2x - 1 \) captures a line where for every increase in \( x \) by one unit, \( y \) increases by two units due to the slope of 2.

Slope-Intercept Form

The slope-intercept form \( y = mx + b \) is a convenient way to write the equation of a straight line. It immediately reveals two important details:

• **The slope \( m \):** This value indicates how much \( y \) changes as \( x \) changes. A slope of 2 means that for every one unit increase in \( x \), \( y \) increases by two units.
• **The y-intercept \( b \):** This is the point where the line crosses the y-axis. In our case, it's -1, which tells us that when \( x = 0 \), \( y = -1 \).

To graph using this form, you start at the y-intercept and use the slope to find other points. A positive slope means the line rises from left to right, while a negative slope means it falls. The slope-intercept form makes it easy to visualize the graph even before drawing it.

Graphing Techniques

Graphing inequalities involves some extra steps compared to graphing equations. Here's a simple strategy:

• **Plotting the line:** Start with the associated linear equation \( y = mx + b \). This will act as a boundary line. For \( y > 2x - 1 \), you'll graph \( y = 2x - 1 \) as a dashed line because the inequality does not include equal to.
• **Choosing the correct line type:** Use a solid line if the inequality is \( \geq \) or \( \leq \). Use a dashed line if the inequality is \( > \) or \( < \).
• **Shading the region:** Determine which side of the line to shade. Substitute a test point (such as (0,0) if it's not on the line) into the inequality. If it satisfies the inequality, shade that side. For \( y > 2x - 1 \), above the line is shaded as those y-values are greater.

These steps ensure that you accurately represent the solution set of the inequality. By graphing inequalities this way, you can visually understand which values satisfy the inequality condition.