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Chapter 8: Problem 19

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{l}|x| \geq 2 \\\|y|<3\end{array}\right.$$

Short Answer

Expert verified
The graph consists of two vertical strips, each bound horizontally between \(y = -3\) and \(y = 3\), and extending indefinitely left and right outside \(x = -2\) and \(x = 2\).

Step by step solution

01

Recognize Absolute Value Functions

The inequalities involved here are based on absolute values. Absolute value inequalities determine a range of values rather than specific points, and the solutions include constraints on both x and y axes.
02

Solve the Inequality |x| 鈮� 2

For the inequality \(|x| \geq 2\), it means that the value of \(x\) can be less than or equal to \(-2\) or greater than or equal to \(2\). In other words, the solution is \(x \leq -2\) or \(x \geq 2\). Therefore, the shaded region on the graph will include all \(x\) values less than or equal to \(-2\) and all \(x\) values greater than or equal to \(2\), resulting in two vertical strips on the left and right sides of the \(x\)-axis.
03

Solve the Inequality |y| < 3

For the inequality \(|y| < 3\), \(y\) must be greater than \(-3\) and less than \(3\). Therefore, the solution here is \(-3 < y < 3\). This results in a horizontal strip along the \(y\)-axis, between \(y = -3\) and \(y = 3\), on the graph.
04

Combine Regions for the System of Inequalities

To graph the system of inequalities, first graph \(x \leq -2\) and \(x \geq 2\) as vertical shaded strips and \(-3 < y < 3\) as a horizontal shaded strip. The solution to the system is the region where these shaded areas overlap, which will result in two rectangular strips parallel to the y-axis in the specified \(y\) range.
05

Sketch the Graph

Using an xy-coordinate plane, sketch two vertical lines at \(x = -2\) and \(x = 2\). Shade the area to the left of \(x = -2\) and right of \(x = 2\). Next, sketch two horizontal lines at \(y = -3\) and \(y = 3\). Ensure the shading is between these horizontal lines. The graph should show the overlapping regions resulting from both inequalities.

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Key Concepts

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

Absolute Value Inequalities
Absolute value inequalities involve expressions like \(|x| \geq 2\) or \(|y| < 3\). The absolute value of a number is its distance from zero on a number line, regardless of direction. It looks like a V-shaped graph when plotted.
  • For \(|x| \geq 2\), it defines all numbers whose distance to zero is at least 2. This includes the intervals \(x \leq -2\) and \(x \geq 2\).
  • For \(|y| < 3\), it captures all numbers within a 3-unit range from zero, resulting in \(-3 < y < 3\).
These inequalities describe conditions or ranges rather than pinpoint values. This makes them ideal for depicting scenarios where a range of solutions is possible, rather than a single solution. By solving these inequalities, you can identify regions on the graph where the solutions lie.
System of Inequalities
A system of inequalities is a set of two or more inequalities with the same variables. Solving a system means finding all variable values satisfying all inequalities simultaneously.In the exercise, two inequalities \(|x| \geq 2\) and \(|y| < 3\) are combined.
  • The inequality \(|x| \geq 2\) creates two vertical strips at the ends of the x-axis.
  • The inequality \(|y| < 3\) creates a horizontal band across the y-axis.
Solving the system involves graphing both inequalities and determining their intersection. The solution region on the graph represents all points that satisfy both inequalities.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
  • Each point on the plane is defined by an ordered pair \((x, y)\) representing its position.
  • The plane is divided into four quadrants by the axes, each with varying signs for x and y values.
When working with inequalities, the coordinate plane allows you to visually represent regions of solutions. It helps in identifying where solutions to the inequalities lie based on their constraints. Understanding the layout of the coordinate plane is essential for plotting and interpreting graphs of inequalities.
Graph Sketching
Graph sketching involves visually representing mathematical functions or inequalities on a coordinate plane.For example, to sketch the graph of \(|x| \geq 2\) and \(|y| < 3\):
  • Draw vertical lines at \(x = -2\) and \(x = 2\).
  • Shade the regions to the left of \(x = -2\) and right of \(x = 2\).
  • Draw horizontal lines at \(y = -3\) and \(y = 3\).
  • Shade the area between these horizontal lines.
Where these shaded areas overlap, you find the solution region for the system of inequalities. This technique helps visualize solutions effectively, revealing insights not easily seen through algebraic manipulation alone.

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Most popular questions from this chapter

Find \(A B\). $$A=\left[\begin{array}{r} 4 \\ -3 \\ 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 5 & 1 \end{array}\right]$$

Verify the identity for $$\boldsymbol{A}=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll}p & \boldsymbol{q} \\\r & s\end{array}\right], \quad \boldsymbol{C}=\left[\begin{array}{ll}\boldsymbol{w} & \boldsymbol{x} \\\\\boldsymbol{y} & z\end{array}\right]$$ and real numbers \(m\) and \(n\). $$A(B+C)=A B+A C$$

Find the partial fraction decomposition. \(\frac{x^{2}+x-6}{\left(x^{2}+1\right)(x-1)}\)

There are three chains, weighing 450 \(610,\) and 950 ounces, each consisting of links of three different sizes. Each chain has 10 small links. The chains also have \(20,30,\) and 40 medium links and \(30,40,\) and 70 large links, respectively. Find the weights of the small, medium, and large links.

Find \(A B\). $$A=\left[\begin{array}{rr} 4 & -2 \\ 0 & 3 \\ -7 & 5 \end{array}\right], \quad B=\left[\begin{array}{l} 3 \\ 4 \end{array}\right]$$
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