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Chapter 6: Problem 8

Graph each inequality. Do not use a calculator. $$x<3+2 y$$

Short Answer

Expert verified
Graph the line from (3, 0) to (5, 1) as a dashed line, and shade the region below the line.

Step by step solution

01

Convert Inequality to Equation

The first step to graphing an inequality is to convert it into an equation, as the line will represent the boundary. So, take the inequality \(x < 3 + 2y\) and convert it into the equation \(x = 3 + 2y\).
02

Rearrange the Equation

We rearrange the equation to better understand the structure of the line. \(x = 3 + 2y\) can also be written in terms of \(y\):\[ y = \frac{x - 3}{2} \] This is in slope-intercept form \(y = mx + c\).
03

Determine Points for the Line

Choose values for \(x\) to find corresponding \(y\) values. Suppose \(x = 3\), then \(y = 0\). If \(x = 5\), substituting gives: \(y = 1\) because:\[ y = \frac{5 - 3}{2} = 1 \]. Use these points (3, 0) and (5, 1) to plot the line on a graph.
04

Draw the Boundary Line

Use the points found in the previous step to draw the boundary line on a graph. Since the inequality is \(x < 3 + 2y\), we draw a dashed line to indicate that points on the line are not included within the solution set.
05

Shade the Solution Region

Since the inequality is \(x < 3 + 2y\), we need to determine which side of the line includes solutions to the inequality. Choose a test point not on the line, such as \((0, 0)\). Substitute into the inequality: \(0 < 3 + 2(0)\) gives \(0 < 3\), which is true, so the region towards the origin should be shaded.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a powerful tool in graphing equations and understanding linear relationships. It is typically expressed as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) denotes the y-intercept. This form makes it easy to identify how a line behaves and its position on a coordinate plane.
  • Slope (\( m \)): Indicates the steepness of the line, showing how much \( y \) changes as \( x \) increases by one unit. A positive slope means the line rises, while a negative slope means it falls.
  • Y-intercept (\( c \)): Reveals the point where the line crosses the y-axis. Understanding this helps place the line accurately on the graph.
Converting an equation to slope-intercept form makes plotting straightforward, revealing relationships between variables instantly.
Boundary Line
The boundary line is a crucial element when dealing with inequalities. It represents the line that separates possible solutions.
  • To find the boundary line, you first convert the inequality into an equation.
  • This line can be solid or dashed. In our example, we use a dashed line because the inequality is less than (<), meaning points on the line aren't included in the solution set.
Visualizing the boundary helps identify regions that satisfy given inequalities and effectively partition the graph.
Solution Region
The solution region is where all the answers to an inequality lie. It鈥檚 the part of the graph that either includes or excludes the boundary line.
  • After drawing the boundary line, you determine which side contains solutions.
  • This is done by shading the region that satisfies the inequality, often based on a test point. In our example, the area towards the origin satisfied the condition \( x < 3 + 2y \).
Finding this area helps in visualizing and understanding the inequality's constraints more concretely.
Test Point Method
The test point method is a straightforward way to determine which side of the boundary line satisfies the inequality.
  • Start by choosing a test point that isn鈥檛 on the boundary line to avoid errors. A common choice is the origin, \((0, 0)\), if it isn't on the line.
  • Substitute this point into the inequality. If it makes the inequality true, the side containing this point is the solution region.
  • If it's false, the opposite side of the line holds the solutions.
This method simplifies the process of identifying the correct solution region, ensuring accuracy in graphing and interpreting inequalities.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq \log x\\\ &y \geq|x-2| \end{aligned}$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=|x-1|\\\&y=x^{2}-4\end{aligned}$$

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. The supply of a certain product is related to its price by the equation \(p=\frac{1}{3} q,\) where \(p\) is in dollars and \(q\) is the quantity supplied in hundreds of units. (a) If this product sells for 9 dollars, what quantity will be supplied by the manufacturer? (b) Suppose that consumer demand for the same product decreases as price increases according to the equation \(p=20-\frac{1}{5} q .\) If this product sells for 9 dollars, what quantity will consumers purchase? How does this compare with the quantity being supplied by the manufacturer at this price? (c) On the basis of parts (a) and (b), what should happen to the price? Explain. (d) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

The table shows weight \(W,\) neck size \(N,\) overall length \(L,\) and chest size \(C\) for four bears. $$\begin{array}{|c|c|c|c|} \hline W \text { (pounds) } & N \text { (inches) } & L \text { (inches) } & C \text { (inches) } \\ \hline 125 & 19 & 57.5 & 32 \\\ 316 & 26 & 65 & 42 \\ 436 & 30 & 72 & 48 \\ 514 & 30.5 & 75 & 54 \end{array}$$ A. We can model these data with the equation $$ W=a+b N+c L+d C $$ where \(a, b, c,\) and \(d\) are constants. To do so, represent a system of linear equations by a \(4 \times 5\) augmented matrix whose solution gives values for \(a, b, c,\) and \(d\) B. Solve the system. Round each value to the nearest thousandth. C. Predict the weight of a bear with \(N=24, L=63\) and \(C=39 .\) Interpret the result.

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.
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