Problem 18 Graph the solution set of the in... [FREE SOLUTION]

Chapter 14: Problem 18

Graph the solution set of the inequality. (GRAPH CANT COPY) $$2 x+2 y \leq-4$$

Short Answer

Expert verified

The line drawn is $y = -x - 2$, which is a downward sloping line that crosses the y-axis at -2. Because of the '$\leq$' sign in the original inequality, the area below the line is shaded.

Step by step solution

Express the inequality in slope-intercept form

The given inequality is $2x + 2y \leq -4$. Divide each term by two to simplify it. This results in $x + y \leq -2$. The inequality is now in a form that looks like the slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Graph the boundary line

To graph the inequality, start by treating the inequality as an equation and drawing that line. So the equation to draw is $y = -x - 2$, which has a slope of -1 and the y-intercept is -2. The line is drawn on the graph with the y intercept at -2, and goes down 1 unit for every unit it moves to the right. Because the original inequality sign was '$\leq$', this is a solid line, because points on the line are included in the solution.

Shade the correct side of the line

Now, decide which side of the line to shade. Since the inequality is $'\leq'$, the solution includes points below the line. Thus, all points below (or to the right of) the line should be shaded.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form

The slope-intercept form is a way to describe linear equations using the pattern:

$y = mx + b$

Here, $m$ represents the slope, which tells us how steep the line is, and $b$ is the y-intercept, which is where the line crosses the y-axis.
To convert an inequality like $2x + 2y \leq -4$ into the slope-intercept form, you first need to simplify.
Divide all parts of the equation by two, resulting in $x + y \leq -2$.
This can be rearranged to get $y \leq -x - 2$, much like the equation $y = -x - 2$.
This format makes it easier to graph by providing direct insight into the line鈥檚 slope and where it touches the y-axis.

Boundary Line

The boundary line is a visual representation of the equation part of your inequality.
In the inequality $y \leq -x - 2$, the boundary line is $y = -x - 2$.
This line acts as a dividing line on the graph marking different solutions to the inequality.

The slope, $-1$, indicates the line moves down one unit vertically for each unit it moves horizontally to the right.
The y-intercept, $-2$, shows where the line crosses the y-axis.

For inequalities such as "$\leq$" or "$\geq$", the boundary line is solid.
A solid line indicates that the points on the line are included in the solution.
Drawing this line accurately is important for later steps like shading.

Inequality Solutions

Inequality solutions are found by determining which set of points satisfy the inequality.
With our inequality $y \leq -x - 2$, we consider both the boundary line and the area around it.
The goal is to identify the region where the inequality holds true.

If the inequality sign is $"<" or "\leq"$, solutions will be found on or below the boundary line.
For $">" or "\geq"$, solutions lie on or above the line.

Always check a point not on the boundary line to confirm the solution set.
This point test ensures the shading represents the correct region.

Shading Graph Regions

Shading is the final step in graphing inequalities, representing all possible solutions.
For the inequality $y \leq -x - 2$, shading is done below the boundary line, as indicated by the $"\leq"$ sign.

Assess which side of the line to shade by selecting a test point.
If this point makes the inequality true, then shade that side.
Commonly used points are $(0, 0)$ or points beyond the boundary line.
If the inequality holds, shade toward that test point.

Remember that shaded areas include all solutions to the inequality, visually clarifying where valid points lie.
This visual demonstration provides a clear understanding of inequality solutions.

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91影视

Graph the solution set of the inequality. (GRAPH CANT COPY) $$2 x+2 y \leq-4$$

Short Answer

Step by step solution

Express the inequality in slope-intercept form

Graph the boundary line

Shade the correct side of the line

Key Concepts

Slope-Intercept Form

Boundary Line

Inequality Solutions

Shading Graph Regions

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