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

Solution:

Step 1: Express in the form $y=mx+b$.

• Step 1.1: Isolate $y$.

  • Step 1.1.1: Multiply $-\frac{1}{3}$ with $x$ to get $y\le -\frac{1}{3}x+4$.

  • Step 1.1.2: Combine the terms involving $x$ to simplify as $y\le -\frac{x}{3}+4$.

• Step 1.2: Rearrange the equation if necessary to maintain the form $y\le -\frac{1}{3}x+4$.

• Step 1.3: Simplify the equation by eliminating any unnecessary parentheses to finalize as $y\le -\frac{1}{3}x+4$.

Step 2: Identify the slope and y-intercept from the slope-intercept form.

• Step 2.1: Determine the values of $m$ (slope) and $b$ (y-intercept) from $y=mx+b$ where $m=-\frac{1}{3}$ and $b=4$.

• Step 2.2: Note that the slope of the line is $m$ and the y-intercept is the point $(0,b)$. Therefore, the slope is $-\frac{1}{3}$ and the y-intercept is at $(0,4)$.

Step 3: Plot the line on a graph.

• Draw a solid line for the equation $y=-\frac{1}{3}x+4$ because the inequality is $\le$.

Step 4: Shade the region.

• Since the inequality is $y\le -\frac{1}{3}x+4$, shade the area below the line to represent all the points that satisfy the inequality.

Knowledge Notes:

To graph an inequality in two variables (like $y\le -\frac{1}{3}x+4$), you need to perform several steps:

1. Linear Equation in 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 gives you the rate of change of the line and where it crosses the y-axis.

2. Slope: The slope ($m$) indicates the steepness of the line and the direction it slants. A negative slope means the line goes downwards from left to right.

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 the Equation: When graphing the equation, you start by plotting the y-intercept and then use the slope to find another point on the line. For instance, a slope of $-\frac{1}{3}$ means you go down 1 unit and right 3 units from the y-intercept to find another point.

5. Solid or Dashed Line: If the inequality is $\le$ or $\ge$, you draw a solid line to indicate that points on the line are included in the solution set. If the inequality is $<$ or $>$, you draw a dashed line to indicate that points on the line are not included in the solution set.

6. Shading the Region: The inequality sign tells you which side of the line to shade. For $y\le mx+b$, you shade below the line; for $y\ge mx+b$, you shade above the line. The shaded region represents all the points that satisfy the inequality.

7. Checking Solutions: You can check if a point is a solution to the inequality by substituting the coordinates into the inequality. If the inequality holds true, the point is part of the solution set.