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

Solution:

Step:1

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

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 $y\le -x-4$.

Step:2

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

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 $-y\le x-4$.

Step:2.1.1.2

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

$$y\ge 4-x$$

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 $y\ge -x+4$.

Step:3

Overlay the two graphs on the same coordinate plane.

$$y\le -x-4$$ $$-x-y\le -4$$

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:

1. 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.

2. Graphing Inequalities: When graphing $y\le mx+b$, the area below the line is shaded, and for $y\ge mx+b$, 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 >).

3. 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.

4. Intersection of Solutions: The solution to a system of inequalities is the region where the shaded areas overlap on the graph.

5. 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.