To solve the given question, we need to graphically represent the system of linear inequalities and determine the correct graph that satisfies both inequalities.

System of inequalities:

1. $x + y < 2$
2. $x + 3y \geq 0$

Step 1: Analyze each inequality

1. For $x + y < 2$:

   • Rewrite as $y < -x + 2$.
   • The boundary line is $y = -x + 2$, which is a straight line with slope $-1$ and y-intercept $2$.
   • The inequality $y < -x + 2$ means we shade the region below the line $y = -x + 2$.

2. For $x + 3y \geq 0$:

   • Rewrite as $y \geq -\frac{1}{3}x$.
   • The boundary line is $y = -\frac{1}{3}x$, which is a straight line with slope $-\frac{1}{3}$ and passes through the origin.
   • The inequality $y \geq -\frac{1}{3}x$ means we shade the region above or on the line $y = -\frac{1}{3}x$.

Step 2: Identify the overlapping region

To satisfy both inequalities:

• The solution lies below the line $y = -x + 2$ and above or on the line $y = -\frac{1}{3}x$.

Step 3: Match with the options

Looking at the provided graphs:

• Option (a) correctly shows the intersection of the shaded regions below $y = -x + 2$ and above $y = -\frac{1}{3}x$.

Correct Answer:

Option (a)

Would you like a detailed explanation of how to graph each inequality or any further clarifications?

Related Questions:

1. How do you find the intersection points of two linear inequalities?
2. What is the difference between a strict inequality ($<, >$) and a non-strict inequality ($\leq, \geq$) in graphing?
3. How do slopes of the lines influence the orientation of the shaded regions?
4. How can you verify which region satisfies an inequality using a test point?
5. How are boundary lines represented differently for strict and non-strict inequalities?

Tip:

Always test a point (like $(0,0)$, unless it lies on a boundary) to confirm the correct region for shading in an inequality.