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
Here are the solutions for the inequalities and graphs: 12. Solve the following inequalities and represent the solution on a graph. a) (1)/(2)x > 4 Step 1: Solve for x. Multiply both sides of the inequality by 2: (1)/(2)x × 2 > 4 × 2 x > 8 Step 2: Represent the solution on a number line. Draw a number line. Place an open circle at x = 8 (because the inequality is strict, x cannot be equal to 8). Draw an arrow extending to the right from the open circle, indicating all values greater than 8. The solution is x > 8. b) 15x + 8 < 48 Step 1: Solve for x. Subtract 8 from both sides of the inequality: 15x + 8 - 8 < 48 - 8 15x < 40 Divide both sides by 15: (15x)/(15) < (40)/(15) x < (8)/(3) Step 2: Represent the solution on a number line. Draw a number line. Place an open circle at x = (8)/(3) (approximately 2.67, because the inequality is strict). Draw an arrow extending to the left from the open circle, indicating all values less than (8)/(3). The solution is x < (8)/(3). 13. Find the inequality represented by each graph. a) Graph 1 Step 1: Identify the type of line. The line is a dashed line, which means the inequality does not include "equal to" (< or >). Step 2: Find the equation of the boundary line. The line passes through points like (0, 0), (1, 1), (2, 2), etc. The slope m = (1-0)/(1-0) = 1. The y-intercept is c = 0. So, the equation of the line is y = x. Step 3: Determine the inequality sign from the shaded region. The region above the dashed line y = x is shaded. Choose a test point in the shaded region, for example, (0, 1). Substitute into y and x: 1 and 0. Since the shaded region is above the line, y values are greater than x values. So, the inequality is y > x. The inequality represented by Graph 1 is y > x. b) Graph 2 Step 1: Identify the type of line. The line is a solid line, which means the inequality includes "equal to" ( or ). Step 2: Find the equation of the boundary line. The line passes through points like (0, 2), (1, 3), (-1, 1), etc. The y-intercept is c = 2. The slope m = (3-2)/(1-0) = 1. So, the equation of the line is y = x + 2. Step 3: Determine the inequality sign from the shaded region. The region below the solid line y = x + 2 is shaded. Choose a test point in the shaded region, for example, (0, 0). Substitute into y and x: 0 and 0. 0 compared to 0 + 2 is 0 2. Since the shaded region is below the line, y values are less than or equal to x+2. So, the inequality is y x + 2. The inequality represented by Graph 2 is y x + 2. That's 2 down. 3 left today — send the next one.