The graph shows the solution to a system of inequalities:
Solid line joining ordered pairs 0, 4 and 9, negative 8. Shade the portion of the graph below the line in the first and fourth quadrants.
Which of the following inequalities is modeled by the graph?
4x + 3y ≤ 12; x ≥ 0
4x + 3y ≥ 12; x ≥ 0
4x − 3y ≤ 12; x ≥ 0
−4x − 3y ≤ 12; x ≥ 0
The correct inequality is: 4x + 3y ≤ 12; x ≥ 0.
We can see from the graph that the line is solid (not dashed), which means the inequality includes the points on the line (equality case). The line has a negative slope (since it goes down from left to right), and we want to shade the portion below the line. This means that values of x and y that satisfy 4x + 3y ≤ 12 are the ones below the line, and since x ≥ 0, we only consider the portion of the graph in the first and fourth quadrants.
Let's check the other answer choices:
- 4x + 3y ≥ 12; x ≥ 0: This inequality would be modeled by a dashed line above the same line in the graph, and we want to shade the portion below. This is not what the graph shows.
- 4x − 3y ≤ 12; x ≥ 0: This inequality would be modeled by a line with a positive slope, and we want to shade below it. This is not what the graph shows.
- −4x − 3y ≤ 12; x ≥ 0: This inequality would be modeled by a line with a negative slope, but we want to shade above it (since the inequality has a negative coefficient for y). This is not what the graph shows.
Therefore, the only option that matches the graph is 4x + 3y ≤ 12; x ≥ 0.
The correct inequality that is modeled by the graph is:
4x - 3y ≤ 12; x ≥ 0