Graph y> =(-1/3)x+4

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 $y\ge -\frac{1}{3}x+4$.

  • Step 1.1.2: Rewrite the inequality as $y\ge -\frac{x}{3}+4$.

• Step 1.2: Arrange the terms in the standard form $y=mx+b$.

• Step 1.3: Simplify to remove any parentheses, resulting in $y\ge -\frac{1}{3}x+4$.

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 $y\ge -\frac{1}{3}x+4$.

• 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 $y\ge -\frac{1}{3}x+4$), you need to perform the following steps:

1. 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).

2. 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.

3. Y-Intercept: The y-intercept ($b$) is the point where the line crosses the y-axis. It is represented by the coordinates $(0,b)$.

4. 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 $y\ge mx+b$, you shade above the line. If it is $y\le mx+b$, you shade below the line.

5. 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?.