which is the graph of the linear inequality $y \\geq -x - 3$ ?

Explanation:

Step1: Analyze the inequality type

The inequality is ( y \geq -x - 3 ). For linear inequalities, if the inequality is ( \geq ) or ( \leq ), the boundary line is solid; if it is ( > ) or ( < ), the boundary line is dashed. Here, since it's ( \geq ), the boundary line ( y=-x - 3 ) should be solid. This eliminates the options with dashed lines (the bottom two graphs).

Step2: Determine the region to shade

To find the region, we can test a point. Let's use the origin ((0,0)). Substitute into the inequality: ( 0 \geq - 0 - 3 ), which simplifies to ( 0 \geq - 3 ), a true statement. So the region containing the origin should be shaded.

Now, let's analyze the two remaining top graphs. The line ( y=-x - 3 ) has a slope of (-1) and a y - intercept of (-3). For the first top graph (left - most top), the shaded region is above the line (since when we test ((0,0)), it's in the shaded region of the first top graph). For the second top graph (right - most top), the shaded region is below the line, which would not satisfy the inequality when we test ((0,0)) (because if we substitute ((0,0)) into ( y < -x - 3), we get ( 0 < - 3), which is false). So the correct graph is the one with the solid line and the shaded region containing the origin (the left - most top graph).

Answer:

The graph with the solid red line ( y = -x - 3) and the shaded region above the line (the left - most top graph among the four given graphs).