This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To define the unshaded region R, we need to find the equations of the three lines forming its boundaries and then determine the correct inequality for each. Step 1: Identify the equation for the horizontal line. The horizontal line passes through y=4. The unshaded region R is below or on this line. So, the first inequality is: y 4 Step 2: Identify the equation for the left diagonal line. This line passes through the points (-4, 0) and (-6, 4). First, calculate the slope m: m = (y_2 - y_1)/(x_2 - x_1) = (4 - 0)/(-6 - (-4)) = (4)/(-2) = -2 Using the point-slope form y - y_1 = m(x - x_1) with point (-4, 0): y - 0 = -2(x - (-4)) y = -2(x + 4) y = -2x - 8 The unshaded region R is above or on this line. To verify, take a point in R, e.g., (0,2): 2 -2(0) - 8 2 -8, which is true. So, the second inequality is: y -2x - 8 Step 3: Identify the equation for the right diagonal line. This line passes through the points (-2, 0) and (0, 4). First, calculate the slope m: m = (y_2 - y_1)/(x_2 - x_1) = (4 - 0)/(0 - (-2)) = (4)/(2) = 2 Using the point-slope form y - y_1 = m(x - x_1) with point (-2, 0): y - 0 = 2(x - (-2)) y = 2(x + 2) y = 2x + 4 The unshaded region R is below or on this line. To verify, take a point in R, e.g., (0,2): 2 2(0) + 4 2 4, which is true. So, the third inequality is: y 2x + 4 The three inequalities that define the unshaded region R are: y & 4 \\ y & -2x - 8 \\ y & 2x + 4 That's 2 down. 3 left today — send the next one.