Question

1. in which graph does the shaded region represent the solution to the inequality ( x - y geq 1 )? four graphs with selection circles are shown below the question

Explanation:

Answer:

To determine the correct graph for \( x - y \geq 1 \) (rewritten as \( y \leq x - 1 \)):

1. Rewrite the inequality: \( x - y \geq 1 \) simplifies to \( y \leq x - 1 \).
2. Graph the boundary line: \( y = x - 1 \) is a straight line with slope \( 1 \) and \( y \)-intercept \( -1 \) (or \( x \)-intercept \( 1 \)). The line should be solid (since the inequality is \( \geq \), including the boundary).
3. Determine the shaded region: For \( y \leq x - 1 \), we shade below the line \( y = x - 1 \) (test a point like \( (0,0) \): \( 0 \leq 0 - 1 \) is false, so shade the opposite side of the line from \( (0,0) \), i.e., below \( y = x - 1 \)).

Among the options, the graph with a solid line \( y = x - 1 \) and shading below the line (consistent with \( y \leq x - 1 \)) is the correct one. (Assuming the third graph in the list matches this description, but the key is identifying the solid line and shading below \( y = x - 1 \).)

(Note: If the options are labeled, the correct one is the one where the shaded region is below \( y = x - 1 \) with a solid boundary line.)