In Exercises 1–26, graph each inequality. y≤(1/3)x

Verified step by step guidance

1. Step 1: Start by identifying the equation of the line associated with the inequality. The inequality given is $y\le \frac{1}{3}x$. The equation of the line is $y=\frac{1}{3}x$.
2. Step 2: Graph the line $y=\frac{1}{3}x$. This is a straight line with a slope of $\frac{1}{3}$ and a y-intercept of 0. Begin by plotting the y-intercept at the origin (0,0).
3. Step 3: Use the slope $\frac{1}{3}$ to find another point on the line. From the origin, move up 1 unit and to the right 3 units to plot the next point (3,1).
4. Step 4: Draw a solid line through the points (0,0) and (3,1). The line is solid because the inequality is $\le$, which includes the boundary line.
5. Step 5: Shade the region below the line to represent the solution to the inequality $y\le \frac{1}{3}x$. This shaded area includes all points where the y-value is less than or equal to $\frac{1}{3}x$.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than). Understanding how to interpret and manipulate inequalities is essential for graphing them accurately.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. The equation y = (1/3)x represents a straight line with a slope of 1/3. To graph the inequality y ≤ (1/3)x, one must first graph the line and then determine the region that satisfies the inequality, which is typically below the line.

Shading Regions in Graphs

When graphing inequalities, it is important to shade the appropriate region of the graph to indicate all the solutions. For the inequality y ≤ (1/3)x, the area below the line, including the line itself, is shaded to represent all the points (x, y) that satisfy the inequality. This visual representation helps in understanding the solution set of the inequality.