Graph y< =-x-4 -x-y< =-4

The problem you've been given is to graph a system of inequalities on a coordinate plane. You are asked to represent the region of the plane that satisfies both of the following inequalities simultaneously:

$ y \leq -x - 4 $
$ -x - y \leq -4 $

The inequalities define regions in the Cartesian coordinate system where the relations are true. The first inequality $ y \leq -x - 4 $ represents all the points below or on the line $ y = -x - 4 $, which is a straight line with a negative slope and a y-intercept of -4. The second inequality $ -x - y \leq -4 $ can be rearranged into $ y \geq x - 4 $, which represents all the points above or on the line $ y = x - 4 $, a line with a positive slope and a y-intercept of -4.

The question is essentially asking to find and shade the area on the graph that meets both conditions, indicating where the two shaded regions overlap. This will give you the solution set for the system of inequalities.

$y \leq - x - 4$$- x - y \leq - 4$

Answer

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Solution:

Step:1

Plot the graph of $y \leq -x - 4$.

Step:1.1

Identify the slope and y-intercept using the slope-intercept formula.

Step:1.1.1

The formula for slope-intercept is $y = mx + b$, where $m$ represents the slope and $b$ the y-intercept.

Step:1.1.2

Determine the slope ($m$) and y-intercept ($b$) from the equation $y = mx + b$.

$$m = -1$$ $$b = -4$$

Step:1.1.3

The line's slope is given by $m$, and the y-intercept is at point $b$.

Slope: $-1$ y-intercept: $(0, -4)$

Step:1.2

Draw a solid line for the equation and shade the region below it, as indicated by $y \leq -x - 4$.

Step:2

Sketch the graph of $-x - y \leq -4$.

Step:2.1

Convert the inequality into the form $y = mx + b$.

Step:2.1.1

Isolate $y$ on one side.

Step:2.1.1.1

Add $x$ to both sides to get $-y \leq x - 4$.

Step:2.1.1.2

Multiply the inequality by $-1$ to solve for $y$, remembering to reverse the inequality sign.

$$y \geq 4 - x$$

Step:2.2

Extract the slope and y-intercept from the slope-intercept form.

Step:2.2.1

Recall that the slope-intercept form is given by $y = mx + b$.

Step:2.2.2

From the equation, identify the slope ($m$) and y-intercept ($b$).

$$m = -1$$ $$b = 4$$

Step:2.2.3

The line's slope is $m$, and the y-intercept is at $b$.

Slope: $-1$ y-intercept: $(0, 4)$

Step:2.3

Plot a solid line for the equation and shade the region above it, as indicated by $y \geq -x + 4$.

Step:3

Overlay the two graphs on the same coordinate plane.

$$y \leq -x - 4$$ $$-x - y \leq -4$$

Step:4

Identify the intersection of the shaded regions to find the solution set.

Knowledge Notes:

To solve a system of linear inequalities and graph the solution set, one needs to understand the following concepts:

Slope-Intercept Form: The equation of a line in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Graphing Inequalities: When graphing $y \leq mx + b$, the area below the line is shaded, and for $y \geq mx + b$, the area above the line is shaded. A solid line is used when the inequality includes equality (鈮?or 鈮?, and a dashed line is used when it does not (< or >).
Solving Linear Inequalities: To solve for $y$, one must isolate $y$ on one side of the inequality. When multiplying or dividing by a negative number, the direction of the inequality sign must be reversed.
Intersection of Solutions: The solution to a system of inequalities is the region where the shaded areas overlap on the graph.
Coordinate Plane: A two-dimensional plane where each point is determined by a pair of numerical coordinates, which are the distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.