Final answer:
To solve the system graphically \(\lvert y \rvert \leq \lvert x \rvert\), draw the lines y = x and y = -x on the coordinate plane and shade the region between them, including the lines themselves.
Step-by-step explanation:
Solving the System Graphically
The inequality \(\lvert y \rvert \leq \lvert x \rvert\) represents a region on the coordinate plane where the absolute value of y is less than or equal to the absolute value of x. To solve this graphically, we need to consider four cases based on the combination of x and y being positive or negative:
When x is positive and y is also positive, the inequality y ≤ x is represented by the graph of Line A: y = x and the area below it.
When x is positive, but y is negative, the inequality -y ≤ x turns into Line D: y = -x and the area above it.
When x is negative and y is positive, the inequality y ≤ -x gets represented by Line B: y = -x and the area below it.
When both x and y are negative, -y ≤ -x translates to Line C: y = x and the area above it.
The region of satisfaction for the given inequality is where these areas overlap, which is essentially the region between the two lines, including the lines themselves.
To represent this graph, we draw Line A: y = x and Line B: y = -x and shade the area in between them up to the lines, thus covering both the positive and negative ranges of x and y. It's important to remember when graphing the inequality that the lines will be solid because the inequality includes the ≤ (less than or equal to) sign.