Problem 65
Question
Graph the solution set of each system of inequalities by hand.
$$\begin{array}{r}-2
Step-by-Step Solution
Verified Answer
The solution set is where \(-2 < x < 2\), \(y > 1\), and \(x > y\), represented by a specific bounded region on the coordinate plane.
1Step 1: Understanding the Inequality \(-2 < x < 2\)
This inequality represents a vertical strip on the coordinate plane where the solution for \(x\) is between \(-2\) and \(2\). This is excluding \(x = -2\) and \(x = 2\). We will draw two vertical dashed lines through the points \(x = -2\) and \(x = 2\).
2Step 2: Understanding the Inequality \(y > 1\)
This inequality represents a region above the horizontal line \(y = 1\) (excluding exactly \(y=1\)). We draw a horizontal dashed line at \(y = 1\) and shade the region above this line to indicate the solutions where \(y > 1\).
3Step 3: Understanding the Inequality \(x - y > 0\)
This inequality can be rearranged to \(x > y\). This represents the region above the line \(y = x\) (excluding the line itself). We will draw a dashed line through the origin with a slope of 1 (45 degrees) and shade the region above this line.
4Step 4: Find the Intersection of All Regions
The solution set of the system of inequalities is where all three shaded regions overlap. This region will be bounded by the lines \(x = -2\), \(x = 2\), above the line \(y = 1\), and above the line \(y = x\).
5Step 5: Graph the Solution Set
On the coordinate plane, draw the vertical dashed lines at \(x = -2\) and \(x = 2\), the horizontal dashed line at \(y = 1\), and the dashed line through the origin with a slope of 1 (\(y = x\)). The overlapping region in the desired quadrant is your solution set.
Key Concepts
Coordinate PlaneSolution SetsShading Regions
Coordinate Plane
The coordinate plane is a two-dimensional surface defined by two perpendicular axes: the x-axis and the y-axis. These axes intersect at a point called the origin. The coordinate plane is used in mathematics to graph points, lines, and curves, which helps us visualize relationships and solutions.
- The x-axis is the horizontal number line.
- The y-axis is the vertical number line.
Each point on the plane is defined by an ordered pair \(x, y\), known as coordinates. The first value in the pair, x, tells you the horizontal position of the point, while the second value, y, tells you the vertical position.
To graph a system of inequalities on this plane, you need to understand what each inequality represents and how to properly show this on the x and y axes using lines or curves.
- The x-axis is the horizontal number line.
- The y-axis is the vertical number line.
Each point on the plane is defined by an ordered pair \(x, y\), known as coordinates. The first value in the pair, x, tells you the horizontal position of the point, while the second value, y, tells you the vertical position.
To graph a system of inequalities on this plane, you need to understand what each inequality represents and how to properly show this on the x and y axes using lines or curves.
Solution Sets
The term 'solution set' refers to all the possible solutions that satisfy a given system of inequalities on the coordinate plane. A system of inequalities can have one solution, multiple solutions, or no solution at all. When graphing, our goal is to find the region where all parts of the system are true simultaneously.
In the context of graphing inequalities, each inequality represents a region on the plane. If you have multiple inequalities, the solution set is the area where these regions overlap.
- For example, in the given system, \(-2 < x < 2\), \(y > 1\), and \(x - y > 0\), you need to find where all these conditions are true.
- To do this, graph each inequality as a separate region and observe where they intersect. This overlapping region is the solution set, where all conditions are met.
In the context of graphing inequalities, each inequality represents a region on the plane. If you have multiple inequalities, the solution set is the area where these regions overlap.
- For example, in the given system, \(-2 < x < 2\), \(y > 1\), and \(x - y > 0\), you need to find where all these conditions are true.
- To do this, graph each inequality as a separate region and observe where they intersect. This overlapping region is the solution set, where all conditions are met.
Shading Regions
Shading regions on the graph is a method to visually represent the solution set of an inequality. Each inequality divides the coordinate plane into two parts: one that satisfies the inequality, and one that does not.
To indicate the solutions for an inequality, shade the side of the boundary line that includes the solutions. Use dashed lines for boundaries that are not included in the solutions (like \(x = -2\) and \(x = 2\) in our problem) and solid lines for those that are included.
- \(y > 1\) is graphed with a horizontal dashed line at \(y = 1\) where the area above is shaded. - \(x - y > 0\) translates to \(x > y\), and is shown with a dashed line going through the origin at a 45-degree angle, shading above this line.
Once all inequalities are plotted, identify and clearly mark the overlapping shaded area which represents the solution set.
Shading makes it easier to visually assess the range of solutions and ensures that further analysis can be done accurately using the graph.
To indicate the solutions for an inequality, shade the side of the boundary line that includes the solutions. Use dashed lines for boundaries that are not included in the solutions (like \(x = -2\) and \(x = 2\) in our problem) and solid lines for those that are included.
- \(y > 1\) is graphed with a horizontal dashed line at \(y = 1\) where the area above is shaded. - \(x - y > 0\) translates to \(x > y\), and is shown with a dashed line going through the origin at a 45-degree angle, shading above this line.
Once all inequalities are plotted, identify and clearly mark the overlapping shaded area which represents the solution set.
Shading makes it easier to visually assess the range of solutions and ensures that further analysis can be done accurately using the graph.
Other exercises in this chapter
Problem 64
Find each matrix product if possible. $$\left[\begin{array}{rrr}-1 & 2 & 0 \\ 0 & 3 & 2 \\ 0 & 1 & 4\end{array}\right]\left[\begin{array}{rrr}2 & -1 & 2 \\ 0 &
View solution Problem 64
Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x-y=3\\\&x^{2}+y^{2}=9\end{aligned}$$
View solution Problem 65
Let \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right],\) where \(a, b,\) and \(c\) are nonzero real numbers. Find \(A^{-1}\).
View solution Problem 65
Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&4 x-y+3 z=-3\\\&3 x+y+z=0\\\&2 x
View solution