Problem 8 Solve the following inequalities... [FREE SOLUTION]

Chapter 6: Problem 8

Solve the following inequalities graphically in two-dimensional plane: $$ 3 y-5 x<30 $$

Short Answer

Expert verified

To solve graphically, plot the dashed line corresponding to the equation $3y - 5x = 30$ by finding its x- and y-intercepts, and then shade the half-plane below this line since the point (0,0) satisfies the inequality.

Step by step solution

- Rewrite the inequality as an equation

To solve the inequality graphically, first rewrite the inequality as an equation by replacing the '<' sign with '=' sign. This gives us the equation of the boundary line. In this case, the inequality is rewritten as: $3y - 5x = 30$.

- Graph the boundary line

Graph the line $3y - 5x = 30$ on a two-dimensional plane. To do this, first solve for y-intercept by setting x to zero, which gives $y = 10$. Then, solve for x-intercept by setting y to zero, which gives $x = -6$. Plot these points and draw the line through them. This line will be a dashed line because the inequality is strict (does not include the boundary).

- Determine which side of the boundary line to shade

Test a point not on the boundary line, such as (0,0). Substitute this point into the original inequality $3y - 5x < 30$. Since $3(0) - 5(0) = 0$, which is less than 30, this point is a solution to the inequality. Therefore, shade the side of the line that includes the point (0,0).

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.

Inequality Graphing

Inequality graphing is a visual way to represent solutions to inequalities on a two-dimensional plane. Unlike equations that show equality, an inequality shows a relationship where one side is not necessarily equal to the other, but rather less than or greater than. To graph inequalities, we begin by representing them as if they were equations, with the understanding that this line, known as the boundary line, allows us to determine the area that represents all possible solutions. For example, given the inequality $3y - 5x < 30$, we would first graph the line $3y - 5x = 30$ and then use additional techniques to find the region that satisfies the inequality.

Boundary Line

The boundary line is a fundamental concept when graphing inequalities. It divides the two-dimensional plane into two regions. For straight line inequalities, this boundary is expressed in the same form as a linear equation. However, depending on whether the inequality symbol is <, >, 鈮�, or 鈮� dictates whether the line is dashed or solid. A dashed line indicates that the points on the line are not part of the solution (strict inequalities), while a solid line indicates they are included (non-strict inequalities). In our example $3y - 5x = 30$, because the original inequality was strict ($<$), the boundary line should be graphed as dashed.

Two-Dimensional Plane

A two-dimensional plane is where we plot the graphs of equations and inequalities, comprising both horizontal (x-axis) and vertical (y-axis) dimensions. Each point on this plane is determined by an ordered pair of numbers (x, y). Understanding this plane is critical for visually solving and interpreting inequalities because it allows us to see the regions where the inequality holds true. Visualizing the exercise $3y - 5x < 30$, each point in the shaded region on the plane satisfies the inequality while those outside do not.

Y-Intercept

The y-intercept is the point at which the graph of an equation or inequality crosses the y-axis. This occurs when the value of x is zero. Determining the y-intercept allows us to have a starting point for drawing the line on a graph. In our exercise, by setting $x = 0$, we find that the y-intercept is $y = 10$. This means that the point (0, 10) is one of the key points we use to graph our boundary line.

X-Intercept

Conversely, the x-intercept is the point where the graph crosses the x-axis, occurring when the value of y is zero. Finding the x-intercept gives us another crucial point for graphing. For the inequality $3y - 5x < 30$, setting $y = 0$ yields an x-intercept of $x = -6$. The point (-6, 0) assists in drawing the boundary line accurately. Plotting both the x-intercept and y-intercept provides two anchor points to accurately draw and therefore define the boundary line on our graph.

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

Solve the following inequalities graphically in two-dimensional plane: $$ 3 y-5 x<30 $$

Short Answer

Step by step solution

- Rewrite the inequality as an equation

- Graph the boundary line

- Determine which side of the boundary line to shade

Key Concepts

Inequality Graphing

Boundary Line

Two-Dimensional Plane

Y-Intercept

X-Intercept

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.