Graph y> =(-1/3)x+4
The question presents a relational expression involving a graph and asks to illustrate the set of points (x, y) on a Cartesian plane that satisfy the inequality y is greater than or equal to (-1/3)x + 4. This involves plotting the line y = (-1/3)x + 4, which represents the boundary of the inequality, and then shading the region above the line to indicate all the points where the y-values are greater than those on the line, adhering to the inequality given.
$y \geq \left(\right. - \frac{1}{3} \left.\right) x + 4$
Answer
Solution:
Step 1: Convert to Slope-Intercept Form
Step 1.1: Isolate $y$ in the inequality.
Step 1.1.1: Multiply $x$ by $-\frac{1}{3}$ to get $y \geq -\frac{1}{3}x + 4$.
Step 1.1.2: Rewrite the inequality as $y \geq -\frac{x}{3} + 4$.
Step 1.2: Arrange the terms in the standard form $y = mx + b$.
Step 1.3: Simplify to remove any parentheses, resulting in $y \geq -\frac{1}{3}x + 4$.
Step 2: Identify Slope and Y-Intercept
Step 2.1: Determine the slope $m$ and y-intercept $b$ from the equation $y = mx + b$.
- Slope $m = -\frac{1}{3}$
- Y-intercept $b = 4$
Step 2.2: Note that the slope of the line is $-\frac{1}{3}$ and the y-intercept is at the point $(0, 4)$.
Step 3: Draw the Graph
Graph the line with a solid border to represent $y \geq -\frac{1}{3}x + 4$.
Since $y$ is greater than or equal to the expression, shade the region above the line.
Knowledge Notes:
To graph an inequality in two variables (like $y \geq -\frac{1}{3}x + 4$), you need to perform the following steps:
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. This form is useful for graphing because it easily shows the rate of change of the line (slope) and where it crosses the y-axis (y-intercept).
Slope: The slope of a line ($m$) is a measure of its steepness. It is calculated as the rise over run between two points on the line. A positive slope means the line is increasing, and a negative slope means the line is decreasing.
Y-Intercept: The y-intercept ($b$) is the point where the line crosses the y-axis. It is represented by the coordinates $(0, b)$.
Graphing Inequalities: When graphing an inequality, you use a solid line if the inequality includes equal to (鈮?or 鈮?, and a dashed line if it does not (> or <). The area above or below the line is shaded to represent all the solutions to the inequality. If the inequality is $y \geq mx + b$, you shade above the line. If it is $y \leq mx + b$, you shade below the line.
Boundary Line: The line itself is called the boundary line because it represents the exact points where the inequality is true when it includes an equal sign (鈮?or 鈮?.
