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Chapter 5: Problem 15

Sketch the graph of the inequality. $$2 y-x \geq 4$$

Short Answer

Expert verified
The graph of the inequality \(2y - x \geq 4\) is a straight line passing through the point (0, 2) with slope of \frac{1}{2}. The shading is above the line.

Step by step solution

01

Rewrite inequality as an equation

The first step is to rewrite the inequality \(2y - x \geq 4\) as an equation: \(2y - x = 4\). You should do this to make the plotting easier, as the line will separate the graph into regions.
02

Rearrange the equation in terms of y

Rearrange the equation \(2y - x = 4\) in terms of \(y\). To do this, add \(x\) to both sides to get \(2y = x + 4\). Then, divide both sides by 2 to isolate \(y\), resulting in \(y = \frac{x}{2} + 2\). This form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, is easier to graph.
03

Plot the line

Next, plot the equation \(y = \frac{x}{2} + 2\) on a graph. Start by plotting the y-intercept, which is \(2\). Then, use the slope \(\frac{1}{2}\) to find another point on the line. Starting from the y-intercept go 1 unit up and 2 units to the right, mark this point. Then draw a line through these two points.
04

Identify the inequality region

Finally, identify the inequality region. Since the inequality is \(\geq\), this means that the solution includes the line and the area that the line covers. If it was \(>\), it would only include the area above the line. In this case, choose a test point, if it satisfies the inequality shade the region containing the test point, otherwise, shade the other region.

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Most popular questions from this chapter

You have a total of $$\$ 500,000$$ that is to be invested in (1) certificates of deposit, (2) municipal bonds, (3) blue-chip stocks, and (4) growth or speculative stocks. How much should be put in each type of investment? The certificates of deposit pay \(3 \%\) simple annual interest, and the municipal bonds pay \(10 \%\) simple annual interest. Over a five-year period, you expect the blue-chip stocks to return \(12 \%\) simple annual interest and the growth stocks to return \(15 \%\) simple annual interest. You want a combined annual return of \(10 \%\) and you also want to have only one-fourth of the portfolio invested in stocks.

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=x+2 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 40 \\ x+y & \leq 30 \\ 2 x+3 y & \leq 65 \end{aligned} $$

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=2.5 x+y\) Constraints: \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \\ 5 x+2 y & \leq 10 \end{aligned}\)

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Reasoning When solving a linear programming problem, you find that the objective function has a maximum value at more than one vertex. Can you assume that there are an infinite number of points that will produce the maximum value? Explain your reasoning.
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