Q. Graph the inequality. \(y < x – 3\)

We will graph the inequality \(y < x – 3\) with a clear, step-by-step procedure.

1. Identify the boundary line.The boundary comes from replacing the inequality sign with equality. The boundary equation is\[ y = x – 3 \]

   This is in slope-intercept form \(y = mx + b\) with slope \(m = 1\) and y-intercept \(b = -3\).

2. Find two points on the boundary so you can draw the line.Find the y-intercept by setting \(x = 0\):\[ y = 0 – 3 = -3 \]

   So one point is \((0,-3)\).

   Find the x-intercept by setting \(y = 0\):

   \[ 0 = x – 3 \]

   \[ x = 3 \]

   So another point is \((3,0)\).

   Draw the straight line through \((0,-3)\) and \((3,0)\). Because the original inequality is strict (<), draw this boundary as a dashed line to indicate points on the line are not included.

3. Decide which side of the boundary to shade.Pick a test point not on the line. A convenient choice is the origin \((0,0)\). Substitute it into the inequality:\[ 0 < 0 – 3 \]

   \[ 0 < -3 \]

   This is false, so the region containing \((0,0)\) is not part of the solution set. Therefore, shade the opposite side of the dashed line.

   For further confirmation, pick a point below the line, for example \((0,-4)\), and test it:

   \[ -4 < 0 – 3 \]

   \[ -4 < -3 \]

   This is true, so points below the line are included in the solution set.

4. Conclusion — how the final graph looks.Graph the dashed line through \((0,-3)\) and \((3,0)\). Shade the half-plane that lies below this line (the side that does not contain the origin). The shaded region together with the dashed boundary visually represents all points \((x,y)\) satisfying\[ y < x – 3. \]

Optional set notation for the solution:

\[ \{(x,y)\in\mathbb{R}^2 : y < x – 3\} \]