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Chapter 7: Problem 28

In Exercises 21-32, use a graphing utility to graph the inequality. \( y \ge -20.74 + 2.66x \)

Short Answer

Expert verified
The graph of the inequality \(y \ge -20.74 + 2.66x\) is a straight line with a slope of 2.66 and a y-intercept of -20.74. The entire region above and including the line is shaded indicating that any point in this region is a solution to the inequality.

Step by step solution

01

Understand The Equation

The first step is understanding the equation. The equation is \(y \ge -20.74 + 2.66x\), which is in the slope-intercept form. Here, the slope (m) is 2.66, which is the number next to 'x', and the y-intercept (c) is -20.74, which is the constant.
02

Plot The Line

The second step is to plot the line accordingly. Since the y-intercept (c) is -20.74, start by placing a point on the y-axis at -20.74. The slope (m) of the line is 2.66. This means for each increase of 1 unit on the x-axis, the line rises by 2.66 units on the y-axis. Draw a straight line that follows this pattern.
03

Shade The Appropriate Region

The inequality symbol in the equation is 'greater than or equal to' sign. This signifies that the solution includes all points above or lying on the drawn line. Therefore, shade the entire region that is above the line on the graph.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a way to express equations of lines. It's represented as \( y = mx + c \). This is a straightforward form that tells you the slope \( m \) and the y-intercept \( c \) directly from the equation.
The slope \( m \) tells us how steep the line is. It indicates the amount \( y \) changes for a one-unit increase in \( x \). The y-intercept \( c \) is where the line crosses the y-axis. It's the value of \( y \) when \( x = 0 \).
  • In \( y \ge -20.74 + 2.66x \), the slope is 2.66, which means the line rises by 2.66 for each step to the right.
  • The y-intercept is -20.74, so the line crosses the y-axis at this point.
Understanding this form helps in plotting the line correctly on a graph.
Graphing Utility
A graphing utility is a tool, often a calculator or software, that assists in graphing functions and inequalities. They're incredibly useful because they can plot complex functions quickly and accurately.
To graph the inequality \( y \ge -20.74 + 2.66x \), you can enter this equation into a graphing calculator or software.
  • First, make sure the calculator is set to the correct mode for inequalities.
  • Next, input the equation keeping the inequality sign.
  • The utility should automatically plot the line and shade the relevant area.
This removes the handwriting errors and visualizes the solution effectively. Plus, it can handle more complex graphs than you might plot by hand.
Inequality Solutions
Inequality solutions for linear graphs require thinking about regions instead of just lines. Consider \( y \ge -20.74 + 2.66x \). This inequality means we want all the points \( y \) that are greater than or equal to \(-20.74 + 2.66x\).
The graph of an inequality includes:
  • The boundary line: Here, it's the line \( y = -20.74 + 2.66x \).
  • The shaded area: This represents all the \( y \) values satisfying the inequality.
  • For \( y \ge -20.74 + 2.66x \), the area is above the line since it includes all the greater or equal \( y \) values.
  • The line itself is part of the solution because of the 'equal to' aspect of 'greater than or equal to'.
Knowing where to shade is crucial: above the line for \( \geq \) or below for \( \leq \). Once graphed, you get a visual representation of all solutions to the inequality.

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Most popular questions from this chapter

Two concentric circles have radii \( x \) and \( y \), where \( y > x \). The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \( y = x \) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

In Exercises 61-70, derive a set of inequalities to describe the region. Parallelogram: vertices at \( (0, 0), (4, 0), (1, 4), (5, 4) \)

In Exercises 7-20, sketch the graph of the inequality. \( 10 \ge y \)

In Exercises 29-34, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. Objective function: \( z = 3x + 4y \) Constraints: \( \hspace{1cm} x \ge 0 \) \( \hspace{1cm} y \ge 0 \) \( x + y \le 1 \) \( 2x + y \le 4 \)

In Exercises 7-20, sketch the graph of the inequality. \( 5x + 3y \ge -15 \)
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