To determine the correct graph that represents the solution set for the given system of inequalities, let’s analyze each inequality separately and identify the region where both inequalities overlap.

Step 1: Rewrite the inequalities in slope-intercept form

The system of inequalities is:

1. $x + 2y < -2$
2. $y - x < 3$

Rewrite Inequality 1: $x + 2y < -2$ Subtract $x$ from both sides: $2y < -x - 2$ Divide by 2: $y < -\frac{1}{2}x - 1$ This inequality represents a line with a slope of $-\frac{1}{2}$ and a y-intercept of $-1$. Since the inequality is $<$, we shade the region below this line.

Rewrite Inequality 2: $y - x < 3$ Add $x$ to both sides: $y < x + 3$ This inequality represents a line with a slope of $1$ and a y-intercept of $3$. Since the inequality is $<$, we shade the region below this line.

Step 2: Determine the Solution Region

The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap.

1. For $y < -\frac{1}{2}x - 1$: Shade below the line with slope $-\frac{1}{2}$ and y-intercept $-1$.
2. For $y < x + 3$: Shade below the line with slope $1$ and y-intercept $3$.

Step 3: Identify the Correct Graph

In the provided options:

• Look for the graph where the shaded region is below both lines, intersecting where both conditions are true.
• Check the orientation of the lines:
  • The line with slope $-\frac{1}{2}$ should go downwards to the right.
  • The line with slope $1$ should go upwards to the right.

Solution

After analyzing, the correct answer appears to be the third graph (from top to bottom), where both inequalities are satisfied and the correct overlapping shaded region is shown.

Would you like a more detailed breakdown of each graph or have any questions?

Related Questions:

1. How do you determine which side of a line to shade for inequalities?
2. What does it mean for a system of inequalities to have no solution?
3. How can you tell if a point satisfies both inequalities in a system?
4. How does the slope of a line affect the direction of shading in inequalities?
5. What steps can you take to graph inequalities without a calculator?

Tip:

For any inequality in the form $y < mx + b$, always shade below the line, and for $y > mx + b$, shade above the line.