Final answer:
The question requests the graphing of a set of linear inequalities, but the given information appears incomplete.
Step-by-step explanation:
The student's question involves graphing a set of linear inequalities and shading the solution region. However, upon examination of the inequalities provided, it seems there might be a typo or some information may have been inadvertently omitted. The inequalities presented (10 + 20 ≤ 140 ; 6 + 18 ≥ 72 ; ≥ 0 ; ≥ 0) are not in the standard form required for graphing. For graphing linear inequalities, we generally need inequalities in forms such as y ≥ mx + b or x + y ≤ k. Nonetheless, if we correct the likely intended form of these inequalities to something that involves variables, for example, x and y, we can then graph them.
To graph linear inequalities:
- Begin by graphing the boundary line of the inequality. This is done by treating the inequality as if it were an equation. For example, if the inequality was y ≥ 2x + 1, you would graph the line y = 2x + 1.
- Determine whether to use a solid or dashed line for the boundary. If the inequality includes equality (e.g., ≤ or ≥), use a solid line. If not (e.g., < or >), use a dashed line.
- Shade the region of the graph that satisfies the inequality. If the inequality is y ≥ mx + b, shade above the boundary line. If the inequality is y < mx + b, shade below the line.
- Repeat this process for each inequality in the system and then find the region where all shaded areas overlap. That is the solution region.
If the student were to provide proper linear inequalities with x and y, I would then be able to graph the system accurately. It's important that the inequalities describe relationships between variables, not just constants.
When the proper inequalities are graphed, the solution region would be the area of the graph where all shaded regions from each inequality overlap, adhering to the conditions that x and y are greater than or equal to zero.