The graphs of the functions $f : x \mapsto |x|-3$ and $g : x \mapsto 2$ are shown in the diagram - Leaving Cert Mathematics - Question (b) - 2012

Point A is where the graph $f(x)$ intersects the line $y = g(x)$.

Setting: $|x| - 3 = 2$ This simplifies to:

a) $x - 3 = 2 \Rightarrow x = 5$
b) $-x - 3 = 2 \Rightarrow x = -5$

Calculating the y-coordinate for $x = 5$: $A = (5, 2).$

Point B is the intersection of the graphs at the coordinates where both functions are equal.

Setting: $|x| - 3 = 2$ Using: $x = 1 \Rightarrow B = (1, 2).$

Point C is determined by solving the equation for $g(x)$. Since $g(x) = 2$ and occurs at $x = 3$: $C = (3, 2).$

Point D is where the graph of $f(x)$ crosses the y-axis: Using $x=0$: $g(0) = 2$ Thus: $D = (0, -3).$

To solve the inequality: $|x - 3| < 2$ This splits into two inequalities: $-2 < x - 3 < 2$ Adding 3 to all sides: $1 < x < 5$ Thus, the solution set is: $1 < x < 5.$