Problem 5 - Algebra and Trigonometry

Question

Graph each inequality. $$y \leq \frac{1}{3} x$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \leq \frac{1}{3} x\) is the line \(y = \frac{1}{3} x\), along with the shading of the area below this line to represent all the y-values that satisfy the inequality.

Step 1: Identifying the line from inequality

From the inequality \(y \leq \frac{1}{3} x\), we can identify that the line we need to draw is \(y = \frac{1}{3} x\). This is a straight line with a slope of 1/3.

Step 2: Plotting the line

we plot the line \(y = \frac{1}{3} x\) as if it were an equality. We can do this by first plotting the y-intercept, which in this case, since there's no addition or subtraction on the right side of the equation, is at the origin (0,0). Next, using the slope, we find that for each step of 3 units to the right (positive x direction), we also step 1 unit up (positive y direction). We continue doing this to plot several points and then connect these points with a line.

Step 3: Shading the area

Since the given inequality is \(y \leq \frac{1}{3} x\), this means that we are looking at the set of y-values that are less than or equal to the line \(y = \frac{1}{3} x\). We thus shade the area below the line to represent all such values of y.