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Chapter 5: Problem 84

Describe the solution set to the system of inequalities. \(x \geq 0, y \geq 0, x \leq 1, y \leq 1\)

Short Answer

Expert verified
The solution set is the rectangular region in the first quadrant bounded by \(0 \le x \le 1 \) and \(0 \le y \le 1 \).

Step by step solution

01

- Understand the Inequalities

There are four inequalities given: 1) \( x \, \geq \, 0 \) 2) \( y \, \geq \, 0 \) 3) \( x \, \leq \, 1 \) 4) \( y \, \leq \, 1 \). These inequalities will determine the valid region for the solution set.
02

- Plot the Individual Inequalities on a Coordinate Plane

On the coordinate plane, plot the vertical line \( x = 0 \) and shade the region to the right of this line. Then, plot the vertical line \( x = 1 \) and shade the region to the left of this line. Next, plot the horizontal line \( y = 0 \) and shade the region above this line. Finally, plot the horizontal line \( y = 1 \) and shade the region below this line.
03

- Identify the Intersection of All Regions

The intersection of all the shaded regions from Step 2 will be the solution set. This will be a rectangular region in the first quadrant, bounded by the lines \( x = 0 \), \( x=1 \), \( y = 0 \), and \( y = 1 \).

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Key Concepts

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

inequality plotting
Plotting inequalities on a graph helps us visualize the regions that satisfy them. Each inequality draws a boundary line on the coordinate plane. For example, the inequality \( x \, \geq \, 0 \) plots the boundary line \( x = 0 \) and shades the region to the right. Thus, for indicators like \( \geq \) or \( \leq \), the boundary line is included in the solution.
To put it simply:
  • For \( x \, \geq \, 0 \), shade right of the line \( x = 0 \).
  • For \( y \, \geq \, 0 \), shade above the line \( y = 0 \).
  • For \( x \, \leq \, 1 \), shade left of the line \( x = 1 \).
  • For \( y \, \leq \, 1 \), shade below the line \( y = 1 \).
Combining these shaded areas on the graph reveals the solution set.
coordinate plane
The coordinate plane is a two-dimensional surface where we plot points, lines, and curves. It consists of an x-axis (horizontal) and a y-axis (vertical). The intersection of these axes, the origin, is at the point (0,0).

In this problem, understanding the coordinate plane helps to plot the inequalities and find their solution set. Each point on this plane represents a pair of coordinates (x,y). For instance:
  • Moving to the right indicates increasing x-values.
  • Moving up indicates increasing y-values.
Thus, plot each inequality to its respective area on the coordinate plane.
solution set
The solution set is the region where all given inequalities overlap. It represents all possible (x,y) pairs that satisfy each inequality simultaneously.

In our problem, after plotting all the inequalities, the shared region is a rectangle within the first quadrant. Specifically, it is bounded by:\[ 0 \leq x \leq 1 \quad \text{and} \quad 0 \leq y \leq 1 \] This means the solution set includes all points from (0,0) to (1,1), outlining a rectangular area. Thus, this area encapsulates the valid (x,y) pairs satisfying all four given inequalities.

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Most popular questions from this chapter

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is \(\$ 240\) and the profit is \(\$ 160 .\) The cost to build a dining room table is \(\$ 320\) and the profit is \(\$ 240\). (See Examples \(2-3)\) Let \(x\) represent the number of kitchen tables produced per month. Let \(y\) represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling \(x\) kitchen tables and \(y\) dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. \- The number of each type of table cannot be negative. \- Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. \- The company can build at most 90 dining room tables. \- The company does not want to exceed a monthly cost of \(\$ 48,000\). c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?

The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration \(A_{0}\) of the drug in the bloodstream at the time of injection. However, the physician knows that after \(3 \mathrm{hr}\), the drug concentration in the blood is \(0.69 \mu \mathrm{g} / \mathrm{dL}\) and after \(4 \mathrm{hr}\), the concentration is \(0.655 \mu \mathrm{g} / \mathrm{dL}\). The model \(A(t)=A_{0} e^{-k t}\) represents the drug concentration \(A(t)\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) in the bloodstream \(t\) hours after injection. The value of \(k\) is a constant related to the rate at which the drug is removed by the body. a. Substitute 0.69 for \(A(t)\) and 3 for \(t\) in the model and write the resulting equation. b. Substitute 0.655 for \(A(t)\) and 4 for \(t\) in the model and write the resulting equation. c. Use the system of equations from parts (a) and (b) to solve for \(k .\) Round to 3 decimal places. d. Use the system of equations from parts (a) and (b) to approximate the initial concentration \(A_{0}\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) at the time of injection. Round to 2 decimal places. e. Determine the concentration of the drug after \(12 \mathrm{hr}\). Round to 2 decimal places.

A system of equations is given in which each equation is written in slope- intercept form. Determine the number of solutions. If the system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. $$ \begin{array}{l} y=\frac{2}{5} x-7 \\ y=\frac{1}{4} x+7 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} \frac{x-2}{8}+\frac{y+1}{2}=-6 \\ \frac{x-2}{2}-\frac{y+1}{4}=12 \end{array} $$

$$ \begin{array}{l} 2(x+y)=2-y \\ 4 x-1=2-5 y \end{array} $$$$ \begin{array}{l} 5(x+y)=9+2 y \\ 6 y-2=10-7 x \end{array} $$
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