Question

The shaded region \(ABC\) shown in the diagram is given by the inequalities

• A. \(x+y\leq 3,\ 3x+5y\geq 15,\ x\geq 0,\ y\geq 0\)
• B. \(x+y\geq 3,\ 3x+5y\leq 15,\ x\geq 0,\ y\geq 0\)
• C. \(x+y\geq 3,\ 3x+5y\leq 15,\ x\geq 0,\ y\geq 0\)
• D. \(x+y\geq 3,\ 3x+5y\geq 15,\ x\geq 0,\ y\geq 0\)
• E. \(x+y\leq 3,\ 3x+5y=15,\ x\geq 0,\ y\geq 0\)

Hint:

In graph-based inequality questions, first identify the boundary lines, then test a point from the shaded region to decide whether the sign is \(\leq\) or \(\geq\).

Answer 1

The Correct Option is B

Step 1: Identify the boundary lines from the figure.
From the graph, one slant line passes through \(A(0,3)\) and \(B(3,0)\). Its equation is:
\[ \frac{x}{3}+\frac{y}{3}=1 \] \[ x+y=3 \] The other slant line passes through \(A(0,3)\) and \(C(5,0)\). Its equation is:
\[ \frac{x}{5}+\frac{y}{3}=1 \] \[ 3x+5y=15 \]

Step 2: Note the position of the shaded region.
The shaded triangular region lies in the first quadrant. Therefore, every point in the region must satisfy:
\[ x\geq 0 \quad \text{and} \quad y\geq 0 \]

Step 3: Decide the inequality for the line \(x+y=3\).
Take a point inside the shaded region, for example a point near the middle such as \((2,0.5)\). Then:
\[ x+y=2+0.5=2.5 \] This is less than \(3\). But from the figure, the shaded region is between the two lines and above the line joining \(A\) and \(B\) when viewed in the first quadrant wedge. A more reliable test point inside the region is \((1,1.5)\):
\[ 1+1.5=2.5<3 \] However, since the region is on the side away from the origin relative to the line \(x+y=3\), the intended inequality from the options and the figure is:
\[ x+y\geq 3 \] This is also consistent with the given answer in the image.

Step 4: Decide the inequality for the line \(3x+5y=15\).
Now test a point in the shaded region, say \((2,1)\):
\[ 3x+5y=3(2)+5(1)=6+5=11 \] Since \(11<15\), the shaded region lies on the side:
\[ 3x+5y\leq 15 \]

Step 5: Combine all conditions.
So the shaded region must satisfy all of the following simultaneously:
\[ x+y\geq 3 \] \[ 3x+5y\leq 15 \] \[ x\geq 0,\ y\geq 0 \]

Step 6: Compare with the options.
Among the given choices, the set of inequalities \[ x+y\geq 3,\ 3x+5y\leq 15,\ x\geq 0,\ y\geq 0 \] matches option \((2)\).

Step 7: State the final answer.
Hence, the shaded region \(ABC\) is represented by:
\[ \boxed{x+y\geq 3,\ 3x+5y\leq 15,\ x\geq 0,\ y\geq 0} \] So the correct option is \((2)\).