/*! 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 23 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 23

Use a graphing utility to graph the inequality. $$y \leq 6-\frac{3}{2} x$$

Short Answer

Expert verified
The line begins at the y-intercept, which is 6, and falls at a rate of -3/2. As the inequality states 'y is less than or equal to', the shading is included below the line, as well as the line itself being solid.

Step by step solution

01

- Understanding the inequality

Analyze the inequality \(y \leq 6-\frac{3}{2} x\), here y can be any number that is less or equal to \(6-\frac{3}{2} x\).
02

- Plotting the equation

Graph the line following the equation \(y = 6-\frac{3}{2} x\). This line will start from the point (0,6) and given the slope is -3/2, it will fall 3 units vertically for every 2 units it moves horizontally.
03

- Shading the area

Because the inequality is \(y \leq 6-\frac{3}{2} x\), inclusive of y being equal to \(6-\frac{3}{2} x\), the line drawn should be a solid line, not a dashed line. Then, because it's 'y is less than or equal to', the area below the line should be shaded to show all points where y is less than \(6-\frac{3}{2} x\).

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 Utilities
Graphing utilities are powerful tools that enable us to visualize equations and inequalities easily. These are computer software or calculator functions that graph equations with just the input of the equation itself.
They help us by:
  • Providing accurate visualizations quickly.
  • Allowing multiple equations or inequalities to be graphed together for comparison.
  • Offering features like zooming and panning to get a detailed view of specific areas.
Understanding how to use these utilities effectively helps in verifying your manual graphing results and can reveal intricate graph details. They are indispensable for complex or time-consuming problems.
Plotting Equations
Plotting equations involves transferring an equation into a visual, coordinate-based form on a graph. To plot the equation of a line like \(y = 6 - \frac{3}{2}x\), follow these steps:
  • Identify the y-intercept, which is the point where the line crosses the y-axis. For this equation, the intercept is at (0,6). This is where you start plotting.
  • Determine the slope, which is -3/2, meaning for every 2 units you move to the right on the x-axis, move 3 units down on the y-axis.
  • Use the slope to find another point and draw the line through these two points.
Plotting equations accurately is fundamental in solving and understanding relationships in algebraic expressions.
Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a line so that it is easy to read and graph. The general form is \(y = mx + b\), where:
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept.
This form makes it straightforward to see how the line is positioned on the coordinate plane:
  • The slope \(m\) shows the rise over the run, indicating how steep the line is.
  • The intercept \(b\) is where the line crosses the y-axis, serving as a starting point for graphing.
Using the slope-intercept form can simplify graphing and helps in understanding how changes to \(m\) and \(b\) affect the graph.
Inequality Shading
Inequality shading is an important aspect of graphing inequalities which helps represent solutions visually. For inequalities like \(y \leq 6 - \frac{3}{2}x\), the graph includes:
  • A solid line for \(y = 6 - \frac{3}{2}x\), because \(y\) can be exactly equal to this line.
  • Shading below the line, which shows all the solutions where \(y\) is less than or equal to \(6 - \frac{3}{2}x\).
When graphing different types of inequalities:
  • "Less than" or "greater than" use dashed lines, as the line itself is not part of the solution.
  • "Less than or equal to" or "greater than or equal to" use solid lines for inclusive solutions.
Shading communicates all potential coordinates that satisfy the inequality, making the graph a valuable visual tool.

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. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(0,0),(2,-2),(4,0)$$

Determine whether the statement is true or false. Justify your answer. Solving a system of equations graphically will always give an exact solution.

A company has budgeted a maximum of \(1,000,000\) for national advertising of an allergy medication. Each minute of television time costs \(100,000\) and each one-page newspaper ad costs \(20,000 .\) Each television ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5 million readers. The company's market research department recommends that at most \(80 \%\) of the advertising budget be spent on television ads. What is the optimal amount that should be spent on each type of ad? What is the optimal total audience?

Solve the system of linear equations and check any solutions algebraically. $$\left\\{\begin{array}{rr} 2 x+4 y+z= & 1 \\ x-2 y-3 z= & 2 \\ x+y-z= & -1 \end{array}\right.$$

Sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. Objective function: \(z=5 x+4 y\) Constraints: $$\begin{aligned}x & \geq 0 \\\y & \geq 0 \\ 2 x+2 y & \geq 10 \\\x+2 y & \geq 6\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.