Problem 36 In Exercises 33-38, sketch the g... [FREE SOLUTION]

Chapter 4: Problem 36

In Exercises 33-38, sketch the graph of the linear inequality. $$ x \leq-2 y+10 $$

Short Answer

Expert verified

The graph of the linear inequality $x \leq -2y + 10$ is represented by a line that starts at the y-intercept of 5 and has a slope of -1/2, with the area above the line shaded to represent all the possible solutions for the inequality. The line is also solid indicating that it also forms part of the solution.

Step by step solution

Set up the equation in slope-intercept form

The first step is to manipulate the problem so it can be put it into the slope-intercept form of $y = mx + c$ where $m$ is the slope and $c$ is the y-intercept. Here, adding 2y to both sides of the inequality gives: $x + 2y \leq 10$. This equates to $y \geq \frac{-x}{2} + 5$.

Graph the line

Now that the equation is in slope-intercept form, identify the y-intercept (the constant), which is 5, and the slope (the coefficient of x), which is -1/2. Start at the y-intercept (0,5) and use the slope to plot the rest of the line. Since slope is rise/run, and the slope is -1/2, from the y-intercept, every time x decreases by 2, y increases by 1. This can be plotted on the graph.

Identify the region

A crucial step is to shade the appropriate region of the inequality. The inequality symbol is 'greater than or equal to', hence the region above the line needs to be shaded. This area of the shading represents all the possible solutions for the inequality.

Indicate the boundary line

Because the inequality was 'greater than or equal to', the line will be solid, indicating that points on the line are included in the solution.

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Key Concepts

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

Graphing Inequalities

Graphing linear inequalities is an essential skill in algebra. It involves plotting a region on a coordinate plane that satisfies an inequality. To graph an inequality, you first need to express it in a form that you can easily work with, usually by isolating the terms involving the variable of interest.

When graphing inequalities:

Identify if the line should be solid or dashed. For "greater than or equal to" (鈮�) or "less than or equal to" (鈮�), use a solid line because points on the line are included in the solution.
For "greater than" (>) or "less than" (<), use a dashed line to indicate that points on the line are not part of the solution.

Understanding the type of line helps determine whether the boundary line should be included in the shaded region.

Slope-Intercept Form

The slope-intercept form of a linear equation is crucial in graphing. This form is given by the equation $y = mx + c$, where $m$ represents the slope, and $c$ indicates the y-intercept.

This format is particularly user-friendly for graphing because:

The y-intercept $(0,c)$ gives you a starting point on the y-axis.
The slope $m$ tells you how to move from that point to find additional points 鈥� up or down and side to side, by the ratio $\frac{rise}{run}$.

Thus, converting an inequality or equation into this form makes plotting and understanding its graph more straightforward.

Y-Intercept

The y-intercept is a key characteristic of a linear function, represented as the point $(0, c)$ where the line crosses the y-axis. Understanding this concept helps to plot the starting point of a line when graphing.

In the example inequality, the y-intercept $c = 5$ means you begin plotting the line at the point $(0, 5)$ on the y-axis. From this fixed point, the line is extended according to the slope.

Grasping the role of the y-intercept is vital:

It provides a concrete point to start graphing the line.
It indicates where the line impacts the y-axis without any influence of the x-variable.

Recognizing the y-intercept facilitates more precise graphing and understanding of linear equations.

Shading Regions

In graphing inequalities, shading the correct region is fundamental. After plotting the boundary line for the inequality, determine which side of the line satisfies the inequality by substituting a test point, usually $(0,0)$ unless it is on the line.

For the inequality $y \geq \frac{-x}{2} + 5$, the region above the line is shaded because it satisfies the condition of being greater than or equal to.

A few tips to remember when shading:

Use a test point not on the line to check which side of the boundary satisfies the inequality.
The shading represents all possible solutions that meet the inequality's conditions.
If a point falls within the shaded region, it is a solution to the inequality.

Proper shading visually represents the solutions and aids in clearly understanding the areas that satisfy the inequality.

91影视

In Exercises 33-38, sketch the graph of the linear inequality. $$ x \leq-2 y+10 $$

Short Answer

Step by step solution

Set up the equation in slope-intercept form

Graph the line

Identify the region

Indicate the boundary line

Key Concepts

Graphing Inequalities

Slope-Intercept Form

Y-Intercept

Shading Regions

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