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Chapter 7: Problem 95

Explain how to graph the solution set of a system of inequalities.

Short Answer

Expert verified
The solution set of a system of inequalities is the intersection of the solutions for each inequality, represented by the overlapping region in the graph.

Step by step solution

01

Identify The Inequalities

Start by identifying the inequalities in the system. Write them in slope-intercept form (y = mx + b), for easy graphing.
02

Graph The Inequalities

Graph each inequality one by one on the same graph. When graphing, use a solid line if the inequality includes 'equal to' (鈮� or 鈮�) and a dotted line if it does not (< or >). After drawing the line, shade the area that satisfies the inequality. When y > or y 鈮�, shade above the line. Conversely, when y < or y 鈮�, shade below the line.
03

Find The Solution Set

The solution set for this system will be defined by the intersection of the areas from the different inequalities. This is the region that satisfies all inequalities simultaneously. It is the shaded area common to all inequalities.

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Key Concepts

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

Slope-Intercept Form
Graphing systems of inequalities often begins with converting the inequalities to slope-intercept form, which is written as \( y = mx + b \). This form is convenient because it immediately gives us important details to visualize on a graph. The "m" in the equation represents the slope. This tells us how steep the line is, showing the direction and angle of the line.
On the other hand, the "b" is the y-intercept. This is simply where the line crosses the y-axis.
  • For example, considering \( y = 2x + 3 \), the slope (m) is 2, and the y-intercept (b) is 3.
  • This means the line starts at the point (0, 3) and rises 2 units for every 1 unit it moves horizontally to the right.
Transforming an inequality into this form helps in graphing, ensuring a clear representation of the line before shading the solution sets.
Solid and Dotted Lines
When graphing inequalities, it鈥檚 key to know when to use solid versus dotted lines. The type of line represents the boundary of the inequality and whether the points on the line are included in the solution set.
  • If the inequality is either \( y \geq mx + b \) or \( y \leq mx + b \), use a solid line. This informs us that the solutions include the exact points on the line.
  • If the inequality is \( y > mx + b \) or \( y < mx + b \), use a dotted line. It implies that the points along the line are not part of the solution.
So the type of line is vital as it accurately portrays the relationship implied by the inequality symbol.
Solution Set Shading
Shading is the way to show which side of the line contains the solutions for the inequality. The way to decide on shading involves the directionality from the line itself.
  • For \( y > mx + b \) or \( y \geq mx + b \), shade the region above the line. This means that all points above the line fulfill the inequality.
  • For \( y < mx + b \) or \( y \leq mx + b \), shade below the line, indicating that points below the line meet the inequality conditions.
Once all inequalities are graphed and shaded, the solution set 鈥� or where all shaded areas overlap 鈥� gives us the range of solutions satisfying all conditions of the system. It is the intersection of the shaded areas that denotes the final solution set.

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Most popular questions from this chapter

For thousands of years, gold has been considered one of Earth's most precious metals. One hundred percent pure gold is 24 -karat gold, which is too soft to be made into jewelry. In the United States, most gold jewelry is 14 -karat gold, approximately \(58 \%\) gold. If 18 -karat gold is \(75 \%\) gold and 12-karat gold is \(50 \%\) gold, how much of each should be used to make a 14 -karat gold bracelet weighing 300 grams?

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

Use the exponential growth model, \(A=A_{0} e^{k t},\) to solve this exercise. In \(1975,\) the population of Europe was 679 million. By \(2015,\) the population had grown to 746 million. a. Find an exponential growth function that models the data for 1975 through 2015 b. By which year, to the nearest year, will the European population reach 800 million?

an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(\quad z=3 x+2 y\) Constraints \(\left\\{\begin{array}{l}x=0, y \geq 0 \\ x+y \leq 8 \\ x+y \geq 4\end{array}\right.\)
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