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

Verified step by step guidance

1. Identify the inequality given: $y\le \frac{1}{3}x$. This means we are looking for all points (x, y) where y is less than or equal to one-third of x.
2. Graph the boundary line $y=\frac{1}{3}x$. Since the inequality includes '≤', the boundary line will be solid, indicating points on the line satisfy the inequality.
3. To graph the line, find at least two points. For example, when $x=\; 0$, $y=\; 0$. When $x=\; 3$, $y=\; 1$. Plot these points and draw a straight line through them.
4. Determine which side of the line to shade. Pick a test point not on the line, such as (0,1). Substitute into the inequality: $1\; \le \; (1/3)(0)\; =\; 0$. Since 1 ≤ 0 is false, do not shade the side containing (0,1). Shade the opposite side.
5. The shaded region along with the solid boundary line represents all solutions to the inequality $y\le \frac{1}{3}x$.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the related linear equation and then shading the region that satisfies the inequality. For y ≤ (1/3)x, the boundary line is y = (1/3)x, and the area below or on this line is shaded to represent all solutions.

Slope-Intercept Form

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. In y ≤ (1/3)x, the slope is 1/3 and the y-intercept is 0, which helps in quickly graphing the boundary line by starting at the origin and rising one unit for every three units moved right.

Solid vs. Dashed Boundary Lines

When graphing inequalities, a solid line is used if the inequality includes equality (≤ or ≥), indicating points on the line satisfy the inequality. A dashed line is used for strict inequalities (< or >), meaning points on the line are not included. Here, y ≤ (1/3)x uses a solid line.