/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Graph each system of inequalitie... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 28

Graph each system of inequalities. \(4 x-3 y \leq 0\) \(x+y \leq 5\)

Short Answer

Expert verified
Graph \(y \geq \frac{4}{3}x\) and \(y \leq 5 - x\) and find the overlapping shaded region.

Step by step solution

01

- Graph the First Inequality

Rewrite the inequality in slope-intercept form to easily graph it. Start with the inequality: \[4x - 3y \leq 0\] Solve for \(y\): \[3y \geq 4x\] Divide every term by 3: \[y \geq \frac{4}{3}x\] Graph the line \(y = \frac{4}{3}x\) and shade above the line because the inequality is \(\geq\).
02

- Graph the Second Inequality

Rewrite the second inequality in slope-intercept form: \[x + y \leq 5\] Solve for \(y\): \[y \leq 5 - x\] Graph the line \(y = 5 - x\) and shade below the line because the inequality is \(\leq\).
03

- Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas of the two inequalities overlap. Look for the intersection of the shaded area above the line \(y \geq \frac{4}{3}x\) and below the line \(y \leq 5 - x\).

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.

Graphing Inequalities
Graphing inequalities involves visualizing regions on a graph where the solutions to the inequalities lie. Each inequality divides the plane into two regions: one that satisfies the inequality and one that does not.
Here鈥檚 the step-by-step method to graph inequalities:
  • First, rewrite the inequality in slope-intercept form (if it's not already). This means putting it in the form of \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.
  • Then, graph the corresponding equality (as a line).
  • If the inequality is \(\geq\) or \(\leq\), draw the line as a solid line to include the boundary in the solution.
  • If the inequality is \(>\) or \(<\), draw the line as a dashed line to exclude it from the solution.
  • Finally, shade the region that satisfies the inequality. For \(y \geq\) or \(y >\), shade above the line. For \(y \leq\) or \(y <\), shade below the line.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common forms for graphing lines. It is written as \(y = mx + b\), where:
  • 'm' represents the slope of the line, which indicates how steep the line is. It鈥檚 the ratio of the rise (the change in y) over the run (the change in x).
  • 'b' represents the y-intercept, which is the point where the line crosses the y-axis.
The benefits of using the slope-intercept form when dealing with inequalities include:
  • It simplifies the process of graphing the boundary line.
  • It makes it easier to determine which side of the line to shade.
For example, in the inequality \(4x - 3y \leq 0\), converting it to the slope-intercept form helps us identify the slope and y-intercept, making the graphing process straightforward.
Solution Region
The solution region of a system of inequalities is the area where the shaded parts of all individual inequalities overlap. To find this region:
  • First, graph each inequality on the same coordinate plane.
  • Shade the region that satisfies each inequality.
  • The solution region is where these shaded areas intersect.
In our specific example, we have two inequalities: \(4x - 3y \leq 0\) and \(x + y \leq 5\). Once both are graphed and shaded, the overlapping region demonstrates all the points (x, y) that satisfy both inequalities simultaneously. This region is the graphical representation of the solution set for the system of inequalities.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Use graphical methods on the given constraints to find the indicated optimal value of the given objective function. \(\begin{aligned} 2 x+3 y & \leq 100 \\ 5 x+4 y & \leq 200 \\ x & \geq 10 \\\ y & \geq 20 \\ \text { Maximize } & x+3 y . \end{aligned}\)

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{l} x^{2}+x y-y^{2}=29 \\ x^{2}-y^{2}=24 \end{array} $$

Which one of the following is a description of the graph of the solution set of the following system? \(x^{2}+y^{2}<25\) \(y>-2\) A. All points outside the circle \(x^{2}+y^{2}=25\) and above the line \(y=-2\) B. All points outside the circle \(x^{2}+y^{2}=25\) and below the line \(y=-2\) C. All points inside the circle \(x^{2}+y^{2}=25\) and above the line \(y=-2\) D. All points inside the circle \(x^{2}+y^{2}=25\) and below the line \(y=-2\)

Graph each system of inequalities. \(2 x+5 y<10\) \(x-2 y<4\)

Graph each ellipse. $$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.