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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Here are the solutions for the inequalities in Q2: Q2 a) 2 - 5x 9 + 2x Step 1: Gather terms with x on one side and constant terms on the other side. -5x - 2x 9 - 2 -7x 7 Step 2: Divide both sides by -7. Remember to reverse the inequality sign when dividing by a negative number. (-7x)/(-7) (7)/(-7) x -1 The solution is x -1 Q2 b) |3 + 2t| 6 Step 1: Convert the absolute value inequality into a compound inequality. -6 3 + 2t 6 Step 2: Subtract 3 from all parts of the inequality. -6 - 3 2t 6 - 3 -9 2t 3 Step 3: Divide all parts by 2. (-9)/(2) (2t)/(2) (3)/(2) -(9)/(2) t (3)/(2) The solution is -(9)/(2) t (3)/(2) Q2 c) (x - 1)/(3x + 5) > 0 Step 1: Find the critical points by setting the numerator and denominator to zero. For the numerator: x - 1 = 0 x = 1 For the denominator: 3x + 5 = 0 3x = -5 x = -(5)/(3) Step 2: Test intervals defined by the critical points (-(5)/(3) and 1). • For x < -(5)/(3) (e.g., x = -2): (-2 - 1)/(3(-2) + 5) = (-3)/(-1) = 3 > 0. This interval is a solution. • For -(5)/(3) < x < 1 (e.g., x = 0): (0 - 1)/(3(0) + 5) = (-1)/(5) < 0. This interval is not a solution. • For x > 1 (e.g., x = 2): (2 - 1)/(3(2) + 5) = (1)/(11) > 0. This interval is a solution. The solution is x < -(5)/(3) or x > 1 Q2 d) (3t + 2)^2 > 16 Step 1: Take the square root of both sides. Remember to use the absolute value. sqrt((3t + 2)^2) > sqrt(16) |3t + 2| > 4 Step 2: Convert the absolute value inequality into two separate inequalities. 3t + 2 > 4 or 3t + 2 < -4 Step 3: Solve each inequality for t. For the first inequality: 3t > 4 - 2 3t > 2 t > (2)/(3) For the second inequality: 3t < -4 - 2 3t < -6 t < (-6)/(3) t < -2 The solution is t < -2 or t > (2)/(3) Q2 e) 2x^2 - x - 3 < 0 Step 1: Find the roots of the quadratic equation 2x^2 - x - 3 = 0. We can factor the quadratic: (2x - 3)(x + 1) = 0 Setting each factor to zero gives the roots: 2x - 3 = 0 2x = 3 x = (3)/(2) x + 1 = 0 x = -1 Step 2: Determine the intervals where the quadratic is less than zero. Since the coefficient of x^2 (which is 2) is positive, the parabola opens upwards. This means the quadratic expression 2x^2 - x - 3 is negative between its roots. The roots are x = -1 and x = (3)/(2). The solution is -1 < x < (3)/(2) Drop the next question.