Problem 16 - Thinking Mathematically

Question

Graph each linear inequality. \(y<-\frac{1}{3} x\)

Step-by-Step Solution

Answer

The graphical solution to the inequality \(y< -\frac{1}{3} x\) is a line graph with a dashed line representing the equation \(y=-\frac{1}{3} x\) with the region below the line shaded.

Step 1: Graph the Line

The equation of the line is \(y=-\frac{1}{3} x\). This is in the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m=-\frac{1}{3}\) and \(b=0\). This implies that the line crosses the y-axis at point (0,0) and has a negative slope. Sketch this line on the graph.

Step 2: Identify the Region to Shade

Since the inequality symbol is '<' not '<=', we'll use a dashed line to represent the line. The inequality \(y< -\frac{1}{3} x\) tells us that the y-values are less than \( -\frac{1}{3} x\). This signifies that we should shade the area below the line.

Step 3: Shade the Region

Shade the area below the line you drew in Step 1, i.e., the region where the y-values are less than \( -\frac{1}{3} x\). By doing so, we represent all the possible solutions that satisfy the given inequality. Any point in the shaded area will satisfy the inequality \(y< -\frac{1}{3} x\)

Key Concepts

Linear Inequality

When it comes to understanding algebra, the concept of a linear inequality is foundational. Unlike a linear equation, which shows exact solutions, a linear inequality uses symbols such as <, >, ≤, or ≥ to represent a range of possible solutions. For example, the inequality presented in the exercise, \(y < -\frac{1}{3} x\), indicates that the solution set includes all the points where the y-value is less than the quantity \(-\frac{1}{3}\) times the x-value. So essentially, the graph of a linear inequality like this will cover a whole region of the coordinate plane, not just a single line.

One of the best ways to visualize these concepts is by graphing the inequality on a coordinate plane. We first draw the corresponding linear equation—if it were an equality—and then use shading to show where the inequality holds true. This shaded area represents all the coordinate pairs (x, y) that satisfy the inequality.

Slope-Intercept Form

Understanding the slope-intercept form of a line, \(y = mx + b\), is crucial as it allows you to graph any linear equation or inequality quickly. In this form, \(m\) represents the slope of the line, which indicates the steepness and direction of the line, while \(b\) represents the y-intercept, the point where the line crosses the y-axis.

In the exercise, the equation \(y = -\frac{1}{3}x\) is already in slope-intercept form. Here, the slope is \(-\frac{1}{3}\), suggesting that for every three units we move horizontally to the right, the line will drop down by one unit. Recognizing this pattern helps in graphing the line accurately before addressing the inequality itself.

Y-Intercept

The y-intercept of a line is where it crosses the y-axis. This is visually represented as the starting point of the line when graphed and is labeled with coordinates where x is 0. In our exercise, the line associated with the inequality \(y = -\frac{1}{3}x\) intersects the y-axis where x is 0, which means the y-intercept is at (0, 0).

This point is significant because it serves as an anchor for graphing the entire line—in combination with the slope, the y-intercept determines the exact position of the line on the graph.

Inequality Shading

After graphing the line, the next step in graphing a linear inequality is to determine which side of the line to shade. This shading represents all the solutions to the inequality. The direction of the shading is determined by the inequality symbol itself. For instance, if the symbol is '<' like in our exercise, it means you shade below the line. If it were '>', then you would shade above.

To help decide which side to shade without guessing, pick a test point not on the line, such as (0,0) if the line doesn't pass through the origin, and plug this into the inequality. If the inequality holds, then shade the side containing that point. In our case, with a '<' symbol, we seek where y-values are less than the corresponding y-values on our line, leading to shading below the line. Remember: for strict inequalities (< or >), use a dashed line, and for inclusive inequalities (≤ or ≥), use a solid line to indicate that points on the line itself are included in the solutions.