\( R=\{x, y)=y \geq 2 x+1 \) and \( y \leq x-2\} \)

It looks like you're trying to describe a region \( R \) defined by two inequalities involving \( x \) and \( y \). Let's clarify the inequalities: 1. \( y \geq 2x + 1 \) 2. \( y \leq x - 2 \) To understand the region \( R \), we can analyze these inequalities graphically. ### Step 1: Graph the inequalities 1. **Graph \( y = 2x + 1 \)**: - This is a straight line with a slope of 2 and a y-intercept of 1. The region above this line (where \( y \geq 2x + 1 \)) is included in \( R \). 2. **Graph \( y = x - 2 \)**: - This is another straight line with a slope of 1 and a y-intercept of -2. The region below this line (where \( y \leq x - 2 \)) is included in \( R \). ### Step 2: Find the intersection points To find the vertices of the region defined by these inequalities, we can set the two equations equal to each other: \[ 2x + 1 = x - 2 \] Solving for \( x \): \[ 2x - x = -2 - 1 \\ x = -3 \] Now, substitute \( x = -3 \) back into either equation to find \( y \): \[ y = 2(-3) + 1 = -6 + 1 = -5 \] So, the intersection point is \( (-3, -5) \). ### Step 3: Determine the feasible region Now we need to determine where the two inequalities overlap: - The region above the line \( y = 2x + 1 \) includes points where \( y \) is greater than or equal to \( 2x + 1 \). - The region below the line \( y = x - 2 \) includes points where \( y \) is less than or equal to \( x - 2 \). ### Step 4: Analyze the region To find the feasible region, we can test points in the different regions defined by the lines: 1. **Above \( y = 2x + 1 \)**: Choose a point like \( (0, 2) \): - \( 2 \geq 2(0) + 1 \) → True - \( 2 \leq 0 - 2 \) → False - So, \( (0, 2) \) is not in \( R \). 2. **Below \( y = x - 2 \)**: Choose a point like \( (0, -1) \): - \( -1 \geq 2(0) + 1 \) → False - \( -1 \leq 0 - 2 \) → True - So, \( (0, -1) \) is not in \( R \). 3. **Check the intersection point \( (-3, -5) \)**: - \( -5 \geq 2(-3) + 1 \) → True - \( -5 \leq -3 - 2 \) → True - So, \( (-3, -5) \) is in \( R \). ### Conclusion The region \( R \) is the area where \( y \) is greater than or equal to \( 2x + 1 \) and less than or equal to \( x - 2 \). This region is bounded by the lines \( y = 2x + 1 \) and \( y = x - 2 \), and the intersection point \( (-3, -5) \) is one of the vertices of the region. To visualize this, you can plot the two lines and shade the appropriate regions to see where they overlap.