Graph y< =(-1/3)x+4
The given problem is asking to visualize the inequality \( y \leq \frac{-1}{3}x + 4 \) on the Cartesian plane. This involves plotting the line \( y = \frac{-1}{3}x + 4 \), which is the boundary of the inequality, and then shading the region that satisfies the inequality (i.e., the area where y is less than or equal to the value given by \( \frac{-1}{3}x + 4 \) for any value of x). The boundary line itself may be included in the shaded region because the inequality is not strict (it allows for \( y \) to be equal to \( \frac{-1}{3}x + 4 \)), which is typically shown by drawing a solid line.
$y \leq \left(\right. - \frac{1}{3} \left.\right) x + 4$
Answer
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 $y \leq -\frac{1}{3}x + 4$.
Step 1.1.2: Combine the terms involving $x$ to simplify as $y \leq -\frac{x}{3} + 4$.
Step 1.2: Rearrange the equation if necessary to maintain the form $y \leq -\frac{1}{3}x + 4$.
Step 1.3: Simplify the equation by eliminating any unnecessary parentheses to finalize as $y \leq -\frac{1}{3}x + 4$.
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 $\leq$.
Step 4: Shade the region.
- Since the inequality is $y \leq -\frac{1}{3}x + 4$, 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 $y \leq -\frac{1}{3}x + 4$), you need to perform several steps:
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.
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.
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 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.
Solid or Dashed Line: If the inequality is $\leq$ or $\geq$, 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.
Shading the Region: The inequality sign tells you which side of the line to shade. For $y \leq mx + b$, you shade below the line; for $y \geq mx + b$, you shade above the line. The shaded region represents all the points that satisfy the inequality.
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.
