Problem 20 - Intermediate Algebra

Question

Graph each inequality. $$ y \leq 2 x $$

Step-by-Step Solution

Answer

The graph includes a solid line through \( y = 2x \) and shading below it.

Step 1: Identify the Slope and Y-Intercept

The given inequality is in the slope-intercept form, which is similar to the equation of a line \( y = mx + b \). Here, \( m = 2 \) is the slope and \( b = 0 \) is the y-intercept. This tells us that the line will pass through the origin (0,0) and have a slope of 2, meaning the line rises 2 units for every 1 unit it runs along the x-axis.

Step 2: Draw the Boundary Line

Since the inequality is \( y \leq 2x \), the boundary line is \( y = 2x \) and it will be solid (not dashed) because the inequality includes equality (\( \leq \)). Start by plotting the y-intercept (0,0), then use the slope to find another point. From (0,0), rise 2 units and run 1 unit to plot the next point at (1,2). Draw the solid line through these points.

Step 3: Shade the Solution Region

The inequality \( y \leq 2x \) indicates that the solutions are the values of \( y \) that are less than or equal to \( 2x \). Shade the area below the line, as this region represents all the points where the y-values are less than or equal to those on the line. This shaded area is the solution region for the inequality.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations. It is primarily written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

• The slope \( m \) describes how steep the line is and the direction it goes. If \( m \, > \, 0 \), the line ascends as it moves from left to right. For negative slopes, \( m \, < \, 0 \), the line descends.
• The y-intercept \( b \) is the point where the line touches the y-axis. In our case, when \( b = 0 \), it means the line crosses the origin, or point (0,0).

In this exercise, the inequality \( y \leq 2x \) has its linear equation part as \( y = 2x \), where \( 2 \) is the slope and \( 0 \) is the y-intercept. This gives us a clear starting point to graph the line, beginning at the origin and rising two units for every unit it moves right.

Boundary Line

In graphing inequalities, the boundary line plays a vital role as it delineates the area of potential solutions. For a greater than or equal to (\( \geq \)) or less than or equal to (\( \leq \)) inequality, the boundary line is solid, signaling that points on the line are included in the solution.

• For inequalities that are strictly greater than (\( > \)) or less than (\( < \)), a dashed line is used, indicating points on the line itself are not under consideration.
• It's vital to first plot the y-intercept on the y-axis and then use the slope to find another point on the line. From the y-intercept, the slope indicates how to move from one point to the next along the line.

For the inequality \( y \leq 2x \), the boundary line is \( y = 2x \). We start at (0,0) and use a solid line because equality (\( \leq \)) is included, meaning every point on this line is a valid solution.

Solution Region

Once the boundary line is drawn, the next step is identifying the solution region, which is the area where all the solutions to the inequality reside. The direction of shading is determined by the inequality symbol.

• If the inequality is of the form \( y < \) or \( y \leq \), you shade below the boundary line. This encompasses all the points where the \( y \)-values are less than those on the line itself.
• If the inequality involves \( y > \) or \( y \geq \), shading occurs above the line because the \( y \)-values here exceed those on the boundary line.

For \( y \leq 2x \), the correct procedure is to shade below the line since \( \leq \) means we seek points where \( y \) is less than or equal to \( 2x \). This shaded area, containing countless solutions, visually represents all potential \( x \) and \( y \) pairs satisfying the inequality.