graph the system of linear inequalities.\n$5x + y \\geq 1$\n$y < 6$
graph the system of linear inequalities.\n$5x + y \\geq 1$\n$y < 6$
Answer
Explanation:
Step1: Analyze (5x + y \geq 1)
Rewrite it as (y \geq -5x + 1). The boundary line is (y=-5x + 1) (solid line, since (\geq)), and we shade above the line (because (y) is greater than or equal to the expression). The slope is (-5), y - intercept is (1).
Step2: Analyze (y < 6)
The boundary line is (y = 6) (dashed line, since (<)), and we shade below the line (because (y) is less than (6)).
Step3: Identify the correct graph
- For (y \geq -5x + 1), the line has a steep negative slope, y - intercept at (1) (close to the origin on the y - axis), and solid. For (y < 6), dashed line at (y = 6) and shade below. The first graph (top one) has a solid line at (y = 6)? Wait, no, (y<6) should have a dashed line. Wait, maybe the first graph: let's check the shading. The region that satisfies both (y \geq -5x + 1) (shade above the dashed line (y=-5x + 1)) and (y < 6) (shade below (y = 6)). The first graph (the upper one) has a solid line at (y = 6)? Wait, no, the inequality (y < 6) should have a dashed line. But maybe in the given options, the first graph (the top - most) has the correct shading for both: the area that is above (y=-5x + 1) (dashed line) and below (y = 6) (solid? Maybe a typo in the graph, but the key is the intersection of the two shaded regions. The first graph's shaded area is the region that is above the dashed line (y=-5x + 1) and below the line (y = 6) (even if the line for (y = 6) is solid, maybe it's a representation). The second graph's shaded area is below (y = 6) and below (y=-5x + 1), which is wrong. So the correct graph is the first one (the upper graph among the two shown).
Answer:
The correct graph is the first (upper) graph, which shows the region that is above the line (y=-5x + 1) (dashed) and below the line (y = 6) (with appropriate shading for both inequalities).