Write down four inequalities which fully describe the unshaded region R in the figure S below.

8. Write down four inequalities which fully describe the unshaded region R in the figure S below. Step 1: Identify the equations of the boundary lines. Line L1 (horizontal line): This line passes through y=2. Its equation is y=2. Line L2 (vertical line): This line passes through x=-2. Its equation is x=-2. Line L3 (diagonal line): This line passes through points (0, -1) and (3, 0). The slope m = (0 - (-1))/(3 - 0) = (1)/(3). Using the point-slope form y - y_1 = m(x - x_1) with (3, 0): y - 0 = (1)/(3)(x - 3) y = (1)/(3)x - 1 Multiply by 3: 3y = x - 3, which can be written as x - 3y = 3. Line L4 (diagonal line): This line passes through points (0, 4) and (4, 0). The slope m = (0 - 4)/(4 - 0) = (-4)/(4) = -1. Using the point-slope form y - y_1 = m(x - x_1) with (4, 0): y - 0 = -1(x - 4) y = -x + 4, which can be written as x + y = 4. Step 2: Determine the inequalities for the unshaded region R. For L1 (y=2): The unshaded region R is below or on the line. So, y 2. For L2 (x=-2): The unshaded region R is to the right of or on the line. So, x -2. For L3 (x - 3y = 3): The unshaded region R is above or on the line. Test a point in R, e.g., (0, 2). 0 - 3(2) = -6. Since -6 < 3, and R is above the line, the inequality is x - 3y 3. For L4 (x + y = 4): The unshaded region R is below or on the line. Test a point in R, e.g., (0, 0). 0 + 0 = 0. Since 0 < 4, the inequality is x + y 4. The four inequalities are: 1. y 2 2. x -2 3. x - 3y 3 4. x + y 4 9. A line PQ = 12.5cm. By using another line, divide PQ into nine equal parts. This is a geometric construction problem. Step 1: Draw a line segment PQ of length 12.5 cm. Step 2: From point P, draw a ray PX at any convenient acute angle to PQ. Step 3: Using a compass, mark off 9 equal segments along the ray PX. Label the points P_1, P_2, P_3, , P_9 such that PP_1 = P_1P_2 = = P_8P_9. The length of these segments is arbitrary, but they must be equal to each other. Step 4: Join the last point P_9 to point Q, forming line segment P_9Q. Step 5: From each of the points P_1, P_2, , P_8, draw lines parallel to P_9Q. These parallel lines will intersect the line segment PQ at points that divide PQ into nine equal parts. The points of intersection on PQ will divide PQ into 9 equal segments.