Problem 8 Graph the solution set of the in... [FREE SOLUTION]

Chapter 14: Problem 8

Graph the solution set of the inequality. (GRAPH CANT COPY) $$y>3 x-9$$

Short Answer

Expert verified

The graph of the inequality consists of a dashed line representing $y=3x-9$, with the region above this line shaded to show the values that solve the inequality.

Step by step solution

Graphing the Line

It's necessary to start by graphing the line as if it was an equation, $y=3x-9$. This line has a slope of 3 and a y-intercept of -9. Plot the y-intercept (0,-9) and then use the slope to find another point. The slope 3 means rise over run, implying for every 3 units up, move 1 unit to the right. Thus you get a second point at (1, -6). Draw a line connecting these points.

Shading the region according to the inequality

The inequality $y > 3x - 9$ informs us that the solution will be comprised of the points above the line $y = 3x - 9$. Because it's a strict inequality (no 'or equals to'), we draw a dashed line to signify that points on the line are not included in the solution. Then, shade the region above the line, which stands for all points $y$ that are greater than $3x-9$.

Verification of solution

To ensure the shading is correct, you can take a sample point from the shaded region and substitute the x and y values in the inequality. If it holds true, then the shading is correct. For example, use point (0,0). Substituting these values gives 0 > -9, which is true, validating that the shading is correct.

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Key Concepts

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

Inequality Solutions

When dealing with inequalities in an algebraic context, it's crucial to understand what an inequality represents. Unlike equations that imply exact equality, inequalities express a relationship where one side can be greater or lesser than the other.
In the context of graphing, inequalities such as $ y > 3x - 9 $ represent not just a single line but an entire region of the coordinate plane.
Key points to remember when working with inequality solutions include:

Solution sets are not just a line, but a region, either above or below it, depending on the inequality sign.
A strict inequality (such as $ > $ or $ < $) does not include the line itself, which is why we draw it as a dashed line rather than solid.
To verify a solution, pick a test point from the shaded region to see if it satisfies the inequality. If it does, the shading is correct.

Understanding these points will help in correctly interpreting and graphing inequality solutions.

Graphing Linear Equations

To graph a linear equation like $ y = 3x - 9 $, there are a few simple steps we need to follow.
First, identify the intercepts from the equation. Here, $ y = 3x -9 $ gives a y-intercept at (0, -9), the point where the line crosses the y-axis.
Next, the slope. A slope of 3 means that for every 3 units moved vertically, you move 1 unit horizontally.
Plot the first point using the y-intercept, then apply the slope to find a second point:

Start at (0, -9) as your initial point.
Moving up 3 units and 1 unit to the right (because the slope is 3), you reach (1, -6).

Connect these points with a straight line. This line represents all solutions $ (x, y) $ that satisfy $ y = 3x - 9 $. Understanding how to graph these points is fundamental for continuing with graphing inequalities.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph points, lines, and shaded regions representing inequalities. It consists of two axes:

x-axis: The horizontal axis, where x-values (horizontal) are measured.
y-axis: The vertical axis, representing y-values (vertical).

These axes divide the plane into four quadrants, each serving as a reference for plotting coordinates.
To effectively use the coordinate plane for problems and exercises:

Origin: The point (0,0) where the x and y axes intersect, is a common reference point.
Each point on the plane is represented as an ordered pair $(x, y)$.
Positive or negative values determine direction: positive y or x values move you up or right from the origin; negatives in either direction move you down or left.

Graphing on the coordinate plane involves understanding how these axes and quadrants interact with algebraic problems like linear equations and inequalities.

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91影视

Graph the solution set of the inequality. (GRAPH CANT COPY) $$y>3 x-9$$

Short Answer

Step by step solution

Graphing the Line

Shading the region according to the inequality

Verification of solution

Key Concepts

Inequality Solutions

Graphing Linear Equations

Coordinate Plane

One App. One Place for Learning.

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