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Chapter 5: Problem 47

Graph the solution set. If there is no solution, indicate that the solution set is the empty set. \(y<\frac{1}{2} x-4\) \(y>-2 x+1\)

Short Answer

Expert verified
The solution set is the overlapping region of the shaded areas from both inequalities.

Step by step solution

01

Graph the first inequality

Start by graphing the boundary line for the inequality \( y < \frac{1}{2} x - 4 \). This is a dashed line because the inequality is strict (<). To graph the line itself, choose two values for \( x \, compute corresponding \( y \), and draw the line through these points. Finally, shade below the line since the inequality is less than (\y < \frac{1}{2} x - 4\).
02

Graph the second inequality

Next, graph the boundary line for the inequality \( y > -2 x + 1 \). This is also a dashed line because the inequality is strict (>). Again, choose two values for \( x \, compute corresponding \( y \), and draw the line through these points. Lastly, shade above this line since the inequality is greater than (\y > -2 x + 1\).
03

Identify the solution set

Identify the region where the shaded areas from both inequalities overlap. This overlapping region represents the solution set of the system of inequalities. If there is no overlapping region, then the solution set is empty.

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Key Concepts

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

Solution Set
In mathematics, a solution set is the collection of all possible solutions to a given equation or inequality. For the system of inequalities provided, we need to find all the points that satisfy both inequalities simultaneously. In this particular example, the solution set is the region where the shaded areas from both inequalities overlap. This means any point within this overlapping region will be a solution to both inequalities. If there is no overlapping area, the solution set would be empty.
Boundary Line
A boundary line represents the limit or edge of the graphed region of an inequality. In our exercise:

For the inequality \(y < \frac{1}{2}x - 4\), the boundary line is given by the equation \( y = \frac{1}{2}x - 4 \). This line is dashed because the inequality is strict (\textless). This indicates that points exactly on the line do not satisfy the inequality.

Similarly, for the inequality \(y > -2x + 1\), the boundary line is given by the equation \( y = -2x + 1 \). This line is also dashed, signifying again that points on this line are not included in the solution set.
Overlapping Region
The overlapping region in graphing inequalities is where the shaded areas of two or more inequalities intersect. This intersection is critical because:

  • It represents all the points that satisfy all given inequalities at the same time.
  • In our exercise with \( y < \frac{1}{2}x - 4 \) and \( y > -2x + 1 \), the region below the dashed line of the first inequality must also coincide with the region above the dashed line of the second inequality.
Identifying this overlapping region, as mentioned in the solution step, is how we determine the solution set for the given system of inequalities.
Graphing Inequalities
Graphing inequalities involves several systematic steps that help visualize the solution set:

  • **Graph the Boundary Line:** Start by converting the inequality to an equation to find the boundary line. If the inequality is strict (\textless or \textgreater), use a dashed line. If it is inclusive (\( \textless = or \textgreater = \)), use a solid line.
  • **Choose Points:** Select values for \(x\) and compute corresponding \(y\) values to plot the boundary line accurately.
  • **Shading:** Depending on the inequality sign, shade the region above or below the line. For \(y \textless \text{line}\), shade below. For \(y \textgreater \text{line}\), shade above.
In our exercise, each inequality was graphed, and the overlapping shaded region was identified as the overall solution set.

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Most popular questions from this chapter

Solve the system. $$ \begin{array}{l} \log _{2} x+3 \log _{2} y=6 \\ \log _{2} x-\log _{2} y=2 \end{array} $$

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. $$ \begin{array}{l} y=\frac{1}{2} x+3 \\ y=2 x+\frac{1}{3} \end{array} $$

Let \(x\) represent the number of country songs that Sierra puts on a playlist on her portable media player. Let \(y\) represent the number of rock songs that she puts on the playlist. For parts (a)-(e), write an inequality to represent the given statement. a. Sierra will put at least 6 country songs on the playlist. b. Sierra will put no more than 10 rock songs on the playlist. c. Sierra wants to limit the length of the playlist to at most 20 songs. d. The number of country songs cannot be negative. e. The number of rock songs cannot be negative. f. Graph the solution set to the system of inequalities from parts (a)-(e).

A farmer has 1200 acres of land and plans to plant corn and soybeans. The input cost (cost of seed, fertilizer, herbicide, and insecticide) for 1 acre for each crop is given in the table along with the cost of machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Corn } & \$ 180 & \$ 80 & \$ 120 \\ \hline \text { Soybeans } & \$ 120 & \$ 100 & \$ 100 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 198,000$$ for input costs and a maximum of $$\$ 110,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre were reversed between the two crops (that is, $$\$ 100$$ per acre for corn and $$\$ 120$$ per acre for soybeans), how many acres of each crop should be planted to maximize profit?

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. $$ \begin{array}{l} y=8 x-\frac{1}{2} \\ y=8 x-\frac{1}{2} \end{array} $$
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