Find the inequalities that satisfy the region labeled R in the figure below.

5. Find the inequalities that satisfy the region labeled R in the figure below. Step 1: Find the equation of line L1. Line L1 passes through (4, 0) and (0, 4). The slope m_1 = (4-0)/(0-4) = (4)/(-4) = -1. The y-intercept is c_1 = 4. The equation of L1 is y = -x + 4, which can be rewritten as x + y = 4. The region R is below L1. Testing a point in R, e.g., (2, 1): 2 + 1 = 3 < 4. Since L1 is a dashed line, the inequality is x + y < 4. Step 2: Find the equation of line L2. Line L2 passes through (4, 0) and (0, -2). The slope m_2 = (-2-0)/(0-4) = (-2)/(-4) = (1)/(2). The y-intercept is c_2 = -2. The equation of L2 is y = (1)/(2)x - 2. Multiply by 2: 2y = x - 4, which can be rewritten as x - 2y = 4. The region R is above L2. Testing a point in R, e.g., (2, 1): 2 - 2(1) = 0 < 4. This means the region is x - 2y < 4. If we want the region above the line, we need to consider the form y > (1)/(2)x - 2. So 2y > x - 4, or x - 2y < 4. Since L2 is a dashed line, the inequality is x - 2y < 4. Step 3: Find the equation of line L3. Line L3 is a vertical line passing through x = 0. The region R is to the right of L3. Testing a point in R, e.g., (2, 1): 2 > 0. Since L3 is a dashed line, the inequality is x > 0. The inequalities that satisfy the region R are: x + y < 4 x - 2y < 4 x > 0 6. Rodi walked from Kisian to Otonglo, a distance of 8 km for 1(1)/(2) hours. He then took a motorbike to Rabuor that travel at a speed of 80 km/h and took 15 minutes. From Rabuor, he boarded a car to Kendu Bay, 60 km away and took 45 minutes to arrive in Kendu Bay. Calculate his average velocity for the whole journey. Step 1: Calculate the total distance covered. • Segment 1 (Walk): Distance d_1 = 8 km. • Segment 2 (Motorbike): Speed v_2 = 80 km/h, Time t_2 = 15 minutes = (15)/(60) hours = 0.25 hours. Distance d_2 = v_2 × t_2 = 80 km/h × 0.25 h = 20 km. • Segment 3 (Car): Distance d_3 = 60 km. Total distance D_total = d_1 + d_2 + d_3 = 8 km + 20 km + 60 km = 88 km. Step 2: Calculate the total time taken. • Segment 1 (Walk): Time t_1 = 1(1)/(2) hours = 1.5 hours. • Segment 2 (Motorbike): Time t_2 = 15 minutes = 0.25 hours. • Segment 3 (Car): Time t_3 = 45 minutes = (45)/(60) hours = 0.75 hours. Total time T_total = t_1 + t_2 + t_3 = 1.5 h + 0.25 h + 0.75 h = 2.5 hours. Step 3: Calculate the average velocity. Average velocity is defined as total displacement divided by total time. Assuming the journey is in a straight line, total displacement equals total distance. Average velocity = D_totalT_total Average velocity = 88 km2.5 h Average velocity = 35.2 km/h The average velocity for the whole journey is 35.2 km/h. 7. Simplify the expression ((-4b + 3a)^2 - (3a + 4b)^2)/(12a) Step 1: Rewrite the numerator using the difference of squares formula, X^2 - Y^2 = (X-Y)(X+Y). Let X = (-4b + 3a) and Y = (3a + 4b). X - Y = (-4b + 3a) - (3a + 4b) = -4b + 3a - 3a - 4b = -8b X + Y = (-4b + 3a) + (3a + 4b) = -4b + 3a + 3a + 4b = 6a The numerator becomes (X-Y)(X+Y) = (-8b)(6a) = -48ab. Step 2: Substitute the simplified numerator back into the expression. (-48ab)/(12a) Step 3: Simplify the expression by canceling common terms. Assuming a ≠ 0: (-48ab)/(12a) = (-48b)/(12) = -4b The simplified expression is -4b. *8. A saleslady earns a commission of 3% and 5% for sale of goods up to Kshs. 100,000 and above Kshs. 100,000 respectively. In a certain month, Sarah's total commission was Kshs. 15,