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Chapter 5: Problem 94

What is a solution of a system of linear inequalities?

Short Answer

Expert verified
A solution to a system of linear inequalities is a set of values for the variables that makes every inequality in the system true at the same time. It's commonly represented as a region on a graph.

Step by step solution

01

Understanding Linear Inequalities

A linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols <, >, 鈮�, or 鈮�. It's a way of comparing linear functions.
02

Understanding Systems of Linear Inequalities

A system of linear inequalities is a set of two or more inequalities that all include the same variables. They are systems because all the inequalities must hold true at the same time.
03

Understanding the Solution of a System of Linear Inequalities

The solution of a system of linear inequalities is the set of all possible values for the variables that satisfies every inequality in the system simultaneously. These solutions often form a region of points, such as a half-plane or a bounded region on a graph.
04

Representing Solutions Graphically

Graphically, a solution to a system of linear inequalities is represented as the intersection of the areas represented by each inequality. This region contains all the points that satisfy all inequalities at the same time and hence, represents the solution to the system.

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Most popular questions from this chapter

In \(1978,\) a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{lcc}\hline & \text { Boeing 727 } & \text { Falcon 20 } \\\\\hline \text { Direct Operating cost } & \$ 1400 \text { per hour } & \$ 500 \text { per hour } \\\\\text { Payload } & 42,000 \text { pounds } & 6000 \text { pounds }\end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to \(\$ 35,000 .\) \(\cdot\) Total payload had to be at least \(672,000\) pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$3 x-2 y \geq 6$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$y \leq 4 x+4$$

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Based on the graph. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280\). What are the lot's dimensions?
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