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
You're on a roll — Here are the solutions for the inequalities and how to represent them graphically. 10. Solve the following inequalities and represent the solution on a number line. a) y x - 1 Step 1: Identify the boundary line. The boundary line is given by the equation y = x - 1. Step 2: Find two points on the boundary line to draw it. • If x = 0, then y = 0 - 1 = -1. So, the point is (0, -1). • If y = 0, then 0 = x - 1 x = 1. So, the point is (1, 0). Step 3: Draw the boundary line. Plot the points (0, -1) and (1, 0) on the graph paper. Since the inequality is y x - 1 (which includes "equal to"), draw a solid line connecting these two points. Step 4: Determine the region to shade. Choose a test point not on the line, for example, the origin (0, 0). Substitute (0, 0) into the inequality y x - 1: 0 0 - 1 0 -1 This statement is true. Therefore, shade the region that contains the test point (0, 0). This will be the region above the line y = x - 1. The solution is the shaded region including the solid line y = x - 1. b) 2x - y 4 Step 1: Rearrange the inequality to make y the subject. -y 4 - 2x Multiply both sides by -1 and reverse the inequality sign: y 2x - 4 Step 2: Identify the boundary line. The boundary line is given by the equation y = 2x - 4. Step 3: Find two points on the boundary line to draw it. • If x = 0, then y = 2(0) - 4 = -4. So, the point is (0, -4). • If y = 0, then 0 = 2x - 4 2x = 4 x = 2. So, the point is (2, 0). Step 4: Draw the boundary line. Plot the points (0, -4) and (2, 0) on the graph paper. Since the inequality is y 2x - 4 (which includes "equal to"), draw a solid line connecting these two points. Step 5: Determine the region to shade. Choose a test point not on the line, for example, the origin (0, 0). Substitute (0, 0) into the inequality y 2x - 4: 0 2(0) - 4 0 -4 This statement is true. Therefore, shade the region that contains the test point (0, 0). This will be the region above the line y = 2x - 4. The solution is the shaded region including the solid line y = 2x - 4. Got more? Send 'em!