Problem 2 Graph each inequality. $$x+y \... [FREE SOLUTION]

Chapter 9: Problem 2

Graph each inequality. $$x+y \geq 2$$

Short Answer

Expert verified

To solve and graph this inequality, first convert it to slope-intercept form which gives $y \geq -x + 2$, and then graph a solid line corresponding to $y = -x + 2 $. Shade the region above the line to represent solutions to the inequality.

Step by step solution

Convert to Slope-Intercept Form

To convert to slope-intercept form, isolate 'y': \[ y \geq -x + 2\] Here the slope 'm' is -1 and the y-intercept 'b' is 2.

Graph the Line

Draw a line on the graph that represents the equation $ y = -x + 2 $. Since our inequality includes equals ( $\geq$), the line should be solid to acknowledge that points on the line are solutions to the inequality.

Shade the Region

As 'y' is greater than or equal to 鈥�-x + 2鈥�, you will shade the region above the line including the line itself as our inequality includes equals ( $\geq$).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 fundamental way of writing linear equations, and it is expressed as $ y = mx + b $. This format clearly shows two key features:

$ m $: The slope of the line. It indicates the steepness and direction of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
$ b $: The y-intercept of the line. This is where the line crosses the y-axis.

For the inequality $ x + y \geq 2 $, converting it to the slope-intercept form gives $ y \geq -x + 2 $. In this equation, the slope $ m $ is $-1$, which means the line will descend, while the y-intercept $ b $ is $2$, indicating that the line will cross the y-axis at the point (0, 2). Recognizing these components helps us quickly sketch the line needed to represent the inequality.

Inequality Symbols

Inequality symbols show the relationship between two expressions. They are crucial when graphing because they dictate which part of the graph to shade:

$ < $ or $ > $: These indicate that the line should be dashed. Dashed lines mean points on the line are NOT part of the solution.
$ \leq $ or $ \geq $: Here, the line should be solid because points on the line ARE included in the solution.

In our example, the inequality is $ y \geq -x + 2 $, indicating that we must draw a solid line. This shows that points directly on the line are valid solutions. Understanding these symbols helps you determine whether to include or exclude the boundary line from the solution.

Graphing Techniques

Graphing inequalities involves a few key steps that help visualize solutions on a graph:

First, graph the corresponding equation $ y = -x + 2 $. Since the inequality is $ \geq $, use a solid line.
To shade the correct region, pick a test point not on the line (like the origin (0,0)), and check if it satisfies the inequality.
If true, shade the region containing the test point. If false, shade the opposite side.

For $ y \geq -x + 2 $, if using the test point (0,0), substitute into the inequality: $ 0 \geq -0 + 2 $ simplifies to $ 0 \geq 2 $, which is false. Therefore, shade the area above the line, which doesn鈥檛 include (0,0). Mastering these techniques helps in accurately representing inequalities on graphs.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Graph each inequality. $$x+y \geq 2$$

Short Answer

Step by step solution

Convert to Slope-Intercept Form

Graph the Line

Shade the Region

Key Concepts

Slope-Intercept Form

Inequality Symbols

Graphing Techniques

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Statistics

Logic and Functions

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.