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Chapter 8: Problem 114

Write a system of inequalities whose solution set includes every point in the rectangular coordinate system.

Short Answer

Expert verified
The system of inequalities that includes every point in the rectangular coordinate system is \(x \geq 0\), \(x \leq 0\), \(y \geq 0\), \(y \leq 0\).

Step by step solution

01

Writing the inequalities for positive and negative values of 'x' and 'y'

First, considering the x-axis, since it includes all real numbers we write two inequalities: \(x \geq 0\) and \(x \leq 0\). Both of these inequalities put together cover all values on the x-axis. Similarly, considering the y-axis which also includes all real numbers, we write the inequalities: \(y \geq 0\) and \(y \leq 0\) representing all values on the y-axis.
02

Combine the inequalities

We put together all four inequalities to represent all points in the rectangular coordinate system. So, we find our system of inequalities to be: \(x \geq 0\), \(x \leq 0\), \(y \geq 0\), \(y \leq 0\). This system satisfies the requirement of the question.

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Most popular questions from this chapter

Consider the following equations: \(\left\\{\begin{array}{l}{5 x-2 y-4 z=3} \\ {3 x+3 y+2 z=-3}\end{array}\right.\) Eliminate \(z\) by copying Equation \(1,\) multiplying Equation 2 by \(2,\) and then adding the equations.

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=2 x+3 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {2 x+3 y \leq 12} \end{array}\right. \end{aligned} $$

Verify the identity: $$\frac{1}{\sin x \cos x}-\frac{\cos x}{\sin x}=\tan x$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Even if a linear system has a solution set involving fractions, such as \(\left\\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\\},\) I can use graphs to determine if the solution set is reasonable.

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?
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