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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Question 1(a) Step 1: Separate the compound inequality into two inequalities. The given inequality is -4 < 2x - (1)/(3)(1 - (1)/(2)x) 1. This can be split into: a) -4 < 2x - (1)/(3)(1 - (1)/(2)x) b) 2x - (1)/(3)(1 - (1)/(2)x) 1 Step 2: Solve the first inequality a). -4 < 2x - (1)/(3) + (1)/(6)x -4 < (2 + (1)/(6))x - (1)/(3) -4 < (13)/(6)x - (1)/(3) Add (1)/(3) to both sides: -4 + (1)/(3) < (13)/(6)x -(12)/(3) + (1)/(3) < (13)/(6)x -(11)/(3) < (13)/(6)x Multiply by (6)/(13): -(11)/(3) · (6)/(13) < x -(22)/(13) < x Step 3: Solve the second inequality b). 2x - (1)/(3)(1 - (1)/(2)x) 1 2x - (1)/(3) + (1)/(6)x 1 (2 + (1)/(6))x - (1)/(3) 1 (13)/(6)x - (1)/(3) 1 Add (1)/(3) to both sides: (13)/(6)x 1 + (1)/(3) (13)/(6)x (3)/(3) + (1)/(3) (13)/(6)x (4)/(3) Multiply by (6)/(13): x (4)/(3) · (6)/(13) x (8)/(13) Step 4: Combine the solutions from Step 2 and Step 3. We have x > -(22)/(13) and x (8)/(13). The solution set is -(22)/(13) < x (8)/(13). Step 5: Illustrate the solution on a number line. The interval is (-(22)/(13), (8)/(13)]. On the number line, place an open circle at -(22)/(13) (approximately -1.69) and a closed circle at (8)/(13) (approximately 0.62). Shade the region between these two points. ` <---------------------------------------------------------------------> -2 -22/13 -1 0 8/13 1 <-----o----------------------------------------------------•------------> ` -(22)/(13) < x (8)/(13)