/*! 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 44 Graph the solution set of each s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 44

Graph the solution set of each system of inequalities. $$\begin{array}{c}x \geq 0 \\\x+y \leq 4 \\\2 x+y \leq 5\end{array}$$

Short Answer

Expert verified
The solution set is the region where x 鈮� 0, x + y 鈮� 4, and 2x + y 鈮� 5 overlap.

Step by step solution

01

Graph the boundary line for x 鈮� 0

First, graph the line for the equation x = 0, which is the y-axis. Since the inequality is x 鈮� 0, shade the region to the right of the y-axis.
02

Graph the boundary line for x + y 鈮� 4

Next, graph the line for the equation x + y = 4. To find where this line intersects the x-axis and the y-axis, set y to 0 and solve for x to get (4, 0), and set x to 0 and solve for y to get (0, 4). Plot these points and draw the line. Since the inequality is x + y 鈮� 4, shade the region below this line.
03

Graph the boundary line for 2x + y 鈮� 5

Now, graph the line for the equation 2x + y = 5. To find where this line intersects the x-axis and the y-axis, set y to 0 and solve for x to get (2.5, 0), and set x to 0 and solve for y to get (0, 5). Plot these points and draw the line. Since the inequality is 2x + y 鈮� 5, shade the region below this line.
04

Find the intersection of the shaded regions

Identify the region on the graph where all three shaded areas overlap. This common region is the solution set for the system of inequalities.

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 is a way to visually represent solutions to inequality problems on a coordinate plane.
When graphing inequalities, we use lines to show the boundary of each inequality.
To know where to shade, we look at the direction of the inequality sign.
  • For an inequality such as \( x \geq 0 \), we shade the region where \( x \) is greater than or equal to 0.
  • For \( x + y \leq 4 \), we find the line \( x + y = 4 \) and shade below it.
  • Similarly, for \( 2x + y \leq 5 \), we shade below the line where \( 2x + y = 5 \).
These shaded areas indicate all the possible solutions to each inequality.
Solution Set
The solution set to a system of inequalities is the region where all the shaded areas from each inequality overlap.
To find this, we graph each inequality one by one and shade the relevant area.
  • First, we graph \( x \geq 0 \), shading everything to the right of the y-axis.
  • Next, we graph \( x + y \leq 4 \) and shade below the line where \( x + y = 4 \).
  • Finally, we graph \( 2x + y \leq 5 \) and shade below the line \( 2x + y = 5 \).
The region where all three of these shadings overlap is the solution set for the entire system of inequalities.
Boundary Lines
Boundary lines represent the limits or edges of each inequality on the graph.
They are crucial to understanding where to shade.
  • For \( x \geq 0 \), the boundary line is at \( x = 0 \), which is the y-axis.
  • For \( x + y \leq 4 \), the boundary line is \( x + y = 4 \).
  • For \( 2x + y \leq 5 \), the boundary line is \( 2x + y = 5 \).
By drawing these boundary lines on a coordinate plane and shading the appropriate regions, we can clearly see where the solution to the system of inequalities lies. This overlapping shaded area, defined by the boundary lines, is our solution set.

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

Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((2,3),(-1,0),\) and \((-2,2)\) Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((-2,4),(2,2),\) and \((4,9)\)

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrr} 9 & 1 & 7 \\ 12 & 5 & 2 \\ 11 & 4 & 3 \end{array}\right|$$

Solve each problem. At the Brendan Berger ranch, 6 goats and 5 sheep sell for \(\$ 305,\) while 2 goats and 9 sheep sell for \(\$ 285 .\) Find the cost of a single goat and of a single sheep.

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product when possible. $$A B$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.