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Chapter 8: Problem 70

Graph the solution set of each system of inequalities. $$\left\\{\begin{array}{rr} -\frac{3}{2} x+y \geq & -3 \\ 2 x+y \leq & 4 \\ 2 x+y \geq & -3 \end{array}\right.$$

Short Answer

Expert verified
The solution set is the overlapped region in all three inequalities, which is graphically represented as the common area covered by the individual inequalities.

Step by step solution

01

Graph the First Inequality

Start by graphing the first inequality, '-3/2x + y 鈮� -3'. To do this, treat the inequality as an equality first and graph '-3/2x + y = -3'. This is a linear function, so identify the y-intercept at -3 and with the slope -3/2. Make sure to draw a solid line to represent 'greater than or equal to'. To represent the inequality, shade the region above the line, since the inequality sign is 'greater than or equal to'.
02

Graph the Second Inequality

Next, graph the second inequality, '2x + y 鈮� 4'. When graphing this, treat it first as '2x + y = 4'. This line intercepts the y-axis at 4 and has a slope of -2. Because the inequality sign is 'less than or equal to', draw this line as a solid line, and shade the region below the line.
03

Graph the Third Inequality

Finally, graph the third inequality, '2x + y 鈮� -3'. Start with '2x + y = -3', which intercepts the y-axis at -3 and has a slope of -2. As this is a 'greater than or equal to' inequality, draw a solid line, and shade the region above the line.
04

Finding the Solution Set

The solution set for this system of inequalities is the region where all the shaded areas overlap. Observe this on the graph and mark it.

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Key Concepts

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

Solution Set
When dealing with systems of inequalities, the solution set is the combined region where all the shaded regions from each inequality overlap on the graph.
This region represents all the possible solutions that satisfy each inequality in the system simultaneously.
That means every point in this shaded area will make all the inequalities true.

To determine the solution set:
  • Graph each inequality on the coordinate plane.
  • Identify the line by treating the inequality as an equation (for example, turning \(-\frac{3}{2}x + y \geq -3\) into \(-\frac{3}{2}x + y = -3\)).
  • Determine if the line should be solid or dashed:
    • Solid for \(\leq\) and \(\geq\).
    • Dashed for \(<\) and \(>\).
  • Shade the appropriate region based on the inequality sign.
Finally, look for the overlapping shaded area to find the solution set, which is the region where all conditions are met.
Linear Inequalities
Linear inequalities resemble linear equations but use inequality signs (\(<, \leq, >, \geq\)) instead of an equal sign.
They represent a region in the coordinate plane, not just a line.

For example, the inequality \(-\frac{3}{2}x + y \geq -3\) describes a half-plane above the line \(-\frac{3}{2}x + y = -3\).
Graphing linear inequalities involves a few steps:
  • Convert the inequality to a linear equation to find the boundary line.
  • Decide whether to use a solid or dashed line based on the inequality type.
  • Choose a test point (usually the origin (0,0), if not on the line) to determine which side of the line to shade.
  • Shade the correct region to indicate all possible solutions that satisfy the inequality.
Remember, linear inequalities define areas where the solutions to an inequality lie, limited by a boundary line.
Systems of Inequalities
A system of inequalities consists of two or more inequalities that are considered together.
The goal is to find the solutions that satisfy all inequalities in the system simultaneously.

Working with systems of inequalities:
  • Graph each inequality in the system on the same set of axes.
  • Pay attention to the shading directions and line styles for each inequality.
  • The solution to the system is the area where the shaded regions of all inequalities overlap.
Solving systems of inequalities graphically lets you visually determine where each inequality holds true.
It helps understand complex interactions between multiple conditions, which can be crucial in real-world scenarios.
Graphing also allows for easy identification of boundaries and feasible regions in optimization problems.

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Most popular questions from this chapter

An electronics firm makes a clock radio in two different models: one (model 380 ) with a battery backup feature and the other (model 360 ) without. It takes 1 hour and 15 minutes to manufacture each unit of the model 380 radio, and only 1 hour to manufacture each unit of the model \(360 .\) At least 500 units of the model 360 radio are to be produced. The manufacturer realizes a profit per radio of \(\$ 15\) for the model 380 and only \(\$ 10\) for the model \(360 .\) If at most 2000 hours are to be allocated to the manufacture of the two models combined, how many of each model should be made to maximize the total profit?

Joi and Cheyenne are planning a party for at least 50 people. They are going to serve hot dogs and hamburgers. Each hamburger costs \(\$ 1\) and each hot dog costs \(\$ .50 .\) Joi thinks that each person will eat only one item, either a hot dog or a hamburger. She also estimates that they will need at least 15 hot dogs and at least 20 hamburgers. How many hamburgers and how many hot dogs should Joi and Cheyenne buy if they want to minimize their cost?

For the given matrices \(A, B,\) and \(C,\) evaluate the indicated expression. $$\begin{aligned}&A=\left[\begin{array}{rr}3 & 1 \\\2 & 5 \\\\-2 & 1\end{array}\right] ; \quad B=\left[\begin{array}{rr}-5 & -3 \\\1 & 6 \\\8 & 3\end{array}\right]\\\&C=\left[\begin{array}{rrr}2 & 1 & 1 \\\0 & -1 & 7 \\\3 & 0 & -3\end{array}\right] ; \quad C B+2 A\end{aligned}$$

Find \(A^{2}\) (the product \(A A\) ) and \(A^{3}\) (the prod\(\left.u c t\left(A^{2}\right) A\right)\). $$A=\left[\begin{array}{rr}1 & 1 \\\\-1 & 2\end{array}\right]$$

Decode the message, which was encoded using the matrix \(\left[\begin{array}{rrr}1 & -2 & 3 \\ -2 & 3 & -4 \\ 2 & -4 & 5\end{array}\right]\). $$\left[\begin{array}{r}-5 \\\0 \\\\-11\end{array}\right],\left[\begin{array}{r}20 \\\\-36 \\\38\end{array}\right]$$
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