Problem 63 Graph the solution. $$y+9 x \geq... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 63

Graph the solution. $$y+9 x \geq 3$$

Short Answer

Expert verified

The inequality is graphed as a solid line representing $y = 3 - 9x$, with the area above the line shaded to indicate the solutions to the inequality.

Step by step solution

Isolate y

To start with the solution, first isolate $y$ in terms of $x$ by subtracting $9x$ from both sides of the inequality, this will result in: $y \geq 3 - 9x$.

Identify the type of line to graph

The inequality $y \geq 3 - 9x$ represents a straight line with a slope of $-9$ and a y-intercept of $3$. It's important to notate that because the inequality is greater than or equal to, the line will be solid, to indicate that points on the line are included in the solution.

Graph the line

Draw a solid line that represents $y = 3 - 9x$. This line will pass through the y-intercept at $3$ and it will slope downwards because the slope is negative.

Shade the appropriate area

Since the inequality is $y \geq 3 - 9x$, you need to shade the area above the line, as points in this area satisfy the inequality.

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

When expressing linear equations or inequalities, the slope-intercept form is one of the most useful formats. It allows you to quickly identify key components of a line: the slope and the y-intercept.
The slope-intercept form is written as $ y = mx + b $, where:

$ m $ is the slope, showing the steepness and direction of the line.
$ b $ is the y-intercept, representing where the line crosses the y-axis.

For example, in the inequality $ y \ge 3 - 9x $, it can be related to the slope-intercept form by rearranging as $ y \ge -9x + 3 $.
Here, the slope is $-9$ and the y-intercept is $3$, which directly tells us how the line behaves graphically.

Linear Inequalities

Linear inequalities are similar to linear equations but with an inequality sign (such as <, >, $\leq$, or $\geq$) instead of an equal sign.
They express a range of solutions, not just a single line.

In our example, $y + 9x \ge 3$ becomes $y \ge 3 - 9x$, showing that the solutions include all the y-values above and on the line $y = -9x + 3$.
When graphing, if the inequality is $\geq$ or $\leq$, the line is solid, indicating that points on the line are included in the solution.
"Greater than" or "less than" inequalities, without the "equal to" part, use dashed lines to show positions not part of the solution set.

Understanding these nuances helps visualize and solve inequalities more effectively on a graph.

Graphical Representation

Graphing is a crucial way to visualize and solve linear inequalities. It makes it easier to see which values fit within an inequality's constraints.
For $y \geq 3 - 9x$:

First, draw the line $y = 3 - 9x$. This is done by marking the y-intercept at $3$ and using the slope of $-9$ to determine the line's direction.
Because the inequality includes the equal sign $\geq$, use a solid line, indicating that points on the line satisfy the inequality.
Shade the region above the line since the inequality states $y$ is greater than or equal to part of the line, including all possible legitimate solutions above and on the boundary.

This method provides a clear picture of where the solutions lie in relation to the linear boundary.

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 the solution. $$y+9 x \geq 3$$

Short Answer

Step by step solution

Isolate y

Identify the type of line to graph

Graph the line

Shade the appropriate area

Key Concepts

Slope-Intercept Form

Linear Inequalities

Graphical Representation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Statistics

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.