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.
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let's tackle these inequalities and the linear equation. Q.8.1.1 Solve the following algebraic inequalities: (2x - 1)/(3) 5 Step 1: Multiply both sides by 3. 3 × (2x - 1)/(3) 5 × 3 2x - 1 15 Step 2: Add 1 to both sides. 2x - 1 + 1 15 + 1 2x 16 Step 3: Divide both sides by 2. (2x)/(2) (16)/(2) x 8 The solution is x 8. Q.8.1.2 Solve the following algebraic inequalities: -5x + 3 8 Step 1: Subtract 3 from both sides. -5x + 3 - 3 8 - 3 -5x 5 Step 2: Divide both sides by -5. Remember to reverse the inequality sign when dividing by a negative number. (-5x)/(-5) (5)/(-5) x -1 The solution is x -1. Q.8.1.3 Solve the following algebraic inequalities: 3x + 2 2(x - 1) Step 1: Distribute the 2 on the right side of the inequality. 3x + 2 2x - 2 Step 2: Subtract 2x from both sides. 3x - 2x + 2 2x - 2x - 2 x + 2 -2 Step 3: Subtract 2 from both sides. x + 2 - 2 -2 - 2 x -4 The solution is x -4. Q.9.1. Determine the linear equation for the line that goes through the following coordinates: (12, 5) and (3:) The second coordinate is incomplete. I will assume the second point is (3, 3) for the purpose of solving. If this is not what you intended, please clarify the second coordinate. Given points: (x_1, y_1) = (12, 5) and (x_2, y_2) = (3, 3). Step 1: Calculate the slope (m) of the line. m = (y_2 - y_1)/(x_2 - x_1) m = (3 - 5)/(3 - 12) m = (-2)/(-9) m = (2)/(9) Step 2: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), with one of the points (e.g., (12, 5)) and the calculated slope. y - 5 = (2)/(9)(x - 12) Step 3: Convert the equation to the slope-intercept form (y = mx + c). y - 5 = (2)/(9)x - (2)/(9) × 12 y - 5 = (2)/(9)x - (24)/(9) y - 5 = (2)/(9)x - (8)/(3) y = (2)/(9)x - (8)/(3) + 5 y = (2)/(9)x - (8)/(3) + (15)/(3) y = (2)/(9)x + (7)/(3) The linear equation is y = (2)/(9)x + (7)/(3). What's next?