Question

Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, y):|x|=2, y \in[0,1]\\} $$

Short answer

Two line segments parallel to the y-axis from \(y=0\) to \(y=1\) at \(x=2\) and \(x=-2\).

Step-by-step solution

Interpret the absolute value condition

Start by analyzing the absolute value condition \(|x|=2\). The absolute value is the distance from zero, so \(|x|=2\) means \(x\) is 2 units away from 0, i.e., \(x=2\) or \(x=-2\).

Understand the interval notation

Secondly, consider the condition \(y \in [0,1]\). The square brackets denote that the endpoints are included. Therefore, \(y\) can be any number within the interval from 0 to 1, including 0 and 1.

Sketch the points

Finally, sketch these points on the \(x-y\) plane that meet both of the conditions. Draw a line parallel to the y-axis at \(x=2\) from \(y=0\) to \(y=1\). Do the same at \(x=-2\). These lines represent all the points which satisfy the given conditions.