Sketch the graph of the system [latex]\left\{\begin{array}{l}|x| \leq 2 \\|y|>1\end{array}\right.[/latex]

Using properties of absolute values (listed on page 68), we see that (x, y) is a solution of the system if and only if both of the following conditions are true: (1) [latex]-2 \leq x \leq 2[/latex] (2) [latex]y<-1 \quad \text {or} \quad y>1[/latex] Thus, a point (x, y) on the graph of the system must lie between (or on) the vertical lines x= ± 2 and also either below the horizontal line y=-1 or above the line y=1. The graph is sketched in Figure 8 .

Question 8.3.5: Sketch the graph of the system {|x| ≤ 2 |y|>1...

Precalculus Functions and Graphs [1893897]

Sketch the graph of the system

$\left\{\begin{array}{l}|x| \leq 2 \\|y|>1\end{array}\right.$

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Step 1:

We start by considering the first equation, which is an inequality: -2 ≤ x ≤ 2. This means that x can take on any value between -2 and 2, inclusive.

Step 2:

Next, we look at the second equation, which is also an inequality: y 1. This means that y can be any value less than -1 or any value greater than 1.

Step 3:

Combining the conditions from both equations, we see that for a point (x, y) to be a solution to the system, it must satisfy both conditions: x must be between -2 and 2, and y must be less than -1 or greater than 1.

Step 4:

Visualizing this on a graph, we sketch two vertical lines at x = -2 and x = 2. Any point between or on these lines satisfies the first condition. Then, we draw a horizontal line at y = -1 and another at y = 1. Any point below the line y = -1 or above the line y = 1 satisfies the second condition. The region that satisfies both conditions is the shaded area on the graph.

Final Answer

Using properties of absolute values (listed on page 68), we see that (x, y) is a solution of the system if and only if both of the following conditions are true:

(1) $-2 \leq x \leq 2$
(2) $y<-1 \quad \text {or} \quad y>1$

Thus, a point (x, y) on the graph of the system must lie between (or on) the vertical lines x= ± 2 and also either below the horizontal line y=-1 or above the line y=1. The graph is sketched in Figure 8 .