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 to the problems: 19. Solve the inequality: |2x + 1| > 3 Step 1: Break the absolute value inequality into two separate inequalities. If |u| > a, then u > a or u < -a. So, we have: 2x + 1 > 3 or 2x + 1 < -3 Step 2: Solve the first inequality. 2x + 1 > 3 2x > 3 - 1 2x > 2 x > (2)/(2) x > 1 Step 3: Solve the second inequality. 2x + 1 < -3 2x < -3 - 1 2x < -4 x < (-4)/(2) x < -2 Step 4: Combine the solutions. The solution is x > 1 or x < -2. The correct option is A. A. x > 1 or x < -2 20. If A B and B A, then: Step 1: Understand the definition of a subset. A B means that every element in set A is also an element in set B. B A means that every element in set B is also an element in set A. Step 2: Apply the definitions to the given conditions. If A is a subset of B and B is a subset of A, it implies that both sets contain exactly the same elements. This is the definition of set equality. Step 3: Conclude the relationship between A and B. Therefore, A = B. The correct option is B. B. A = B 21. Rationalize: (1)/(sqrt(5) - 2) Step 1: Identify the conjugate of the denominator. The denominator is sqrt(5) - 2. Its conjugate is sqrt(5) + 2. Step 2: Multiply the numerator and the denominator by the conjugate. (1)/(sqrt(5) - 2) × sqrt(5) + 2sqrt(5) + 2 Step 3: Perform the multiplication. In the numerator: 1 × (sqrt(5) + 2) = sqrt(5) + 2. In the denominator, use the difference of squares formula (a - b)(a + b) = a^2 - b^2: (sqrt(5) - 2)(sqrt(5) + 2) = (sqrt(5))^2 - (2)^2 = 5 - 4 = 1 Step 4: Write the simplified expression. sqrt(5) + 21 = sqrt(5) + 2 The correct option is A. A. sqrt(5) + 2 22. If z = 2 - 2i, find |z| Step 1: Identify the real and imaginary parts of the complex number. For a complex number z = a + bi, a is the real part and b is the imaginary part. Here, z = 2 - 2i, so a = 2 and b = -2. Step 2: Use the formula for the modulus of a complex number. The modulus (or absolute value) of a complex number z = a + bi is given by: |z| = sqrt(a^2 + b^2) Step 3: Substitute the values of a and b into the formula. |z| = sqrt((2)^2 + (-2)^2) Step 4: Calculate the squares and sum them. |z| = sqrt(4 + 4) |z| = sqrt(8) Step 5: Simplify the square root. |z| = sqrt(4 × 2) |z| = sqrt(4) × sqrt(2) |z| = 2sqrt(2) The modulus of z is 2sqrt(2).