Question

Sketch the graph of \(|x+3| \leq y\) in a coordinate plane.

Short answer

To graph \(|x+3| \leq y\) on a coordinate plane, one should draw two solid lines represented by \(y = x+3\) and \(y = -x-3\), then shade the area above these two lines.

Step-by-step solution

Understanding Absolute Values

When you see \(|x+3|\), it means the absolute value of \(x+3\). The absolute value of a number is its distance from zero on the number line, and it is always non-negative. For \(|x+3|\), it represents all x that satisfies \(x+3>=0\) and \(x+3<=0\). Therefore it breaks into two cases: \(x+3\) for \(x>=-3\) and \(-x-3\) for \(x<-3\).

Drawing the Graph

Based on the two cases, you can draw two lines on the graph: one is \(y = x+3\), which starts at \(-3\) on the x-axis and goes upwards to the right; the other is \(y = -x-3\), which also starts from \(-3\) on the x-axis and goes upwards to the left. The area under the lines is the one you are looking for, because you have \(|x+3| \leq y\), which means y values should be above lines \(y = x+3\) and \(y = -x-3\).

Finalizing the Graph

You can finalise your graph by paying attention to one key factor: Since the inequality is not strict (it's '\(\leq\)' not '<'), the lines \(y = x+3\) and \(y = -x-3\) should be represented with solid, not dotted lines, which include the points on the lines. This indicates that the y values on the lines are also solutions to the inequality. Also, shade the area above the two lines to indicate all possible solutions to the inequality.