Graph each inequality. x < 3 + 2y

Verified step by step guidance

1. Rewrite the inequality to isolate \( y \) on one side. Starting with \( x < 3 + 2y \), subtract 3 from both sides to get \( x - 3 < 2y \).
2. Divide both sides of the inequality by 2 to solve for \( y \): \( \frac{x - 3}{2} < y \). This can be rewritten as \( y > \frac{x - 3}{2} \).
3. Identify the boundary line for the inequality, which is \( y = \frac{x - 3}{2} \). Since the inequality is strict (\( > \)), the boundary line will be dashed on the graph.
4. Determine which side of the boundary line to shade. Pick a test point not on the line, such as \( (0,0) \), and substitute into the inequality \( y > \frac{x - 3}{2} \). If the test point satisfies the inequality, shade that side; otherwise, shade the opposite side.
5. Draw the dashed boundary line \( y = \frac{x - 3}{2} \) on the coordinate plane and shade the region above the line (if the test point satisfies the inequality) to represent all solutions to the inequality.

Key Concepts

Inequalities in Two Variables

An inequality involving two variables, like x and y, represents a region in the coordinate plane where the inequality holds true. Instead of a single line, the solution is a shaded area that satisfies the inequality condition.

Graphing Linear Inequalities

To graph a linear inequality, first rewrite it in a form that relates x and y, then graph the boundary line (using equality). Use a solid line if the inequality includes equality (≤ or ≥), or a dashed line if it does not (< or >).

Testing Points to Determine Shading

After drawing the boundary line, select a test point not on the line (commonly (0,0)) to check if it satisfies the inequality. Shade the side of the line where the inequality holds true, representing all solutions.