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: $$2x + 1 > 3 \quad \text{or} \quad 2x + 1 < -3$$ Step 2: Solve the first inequality. $$2x + 1 > 3$$ $$2x > 3 - 1$$ $$2x > 2$$ $$x > \frac{2}{2}$$ $$x > 1$$ Step 3: Solve the second inequality. $$2x + 1 < -3$$ $$2x < -3 - 1$$ $$2x < -4$$ $$x < \frac{-4}{2}$$ $$x < -2$$ Step 4: Combine the solutions. The solution is $x > 1$ or $x < -2$. The correct option is A. $\boxed{\text{A. } x > 1 \text{ or } x < -2}$ 20. If $A \subseteq B$ and $B \subseteq A$, then: Step 1: Understand the definition of a subset. $A \subseteq B$ means that every element in set $A$ is also an element in set $B$. $B \subseteq A$ 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. $\boxed{\text{B. } A = B}$ 21. Rationalize: $\frac{1}{\sqrt{5} - 2}$ 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. $$ \frac{1}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2} $$ Step 3: Perform the multiplication. In the numerator: $1 \times (\sqrt{5} + 2) = \sqrt{5} + 2$. 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. $$ \frac{\sqrt{5} + 2}{1} = \sqrt{5} + 2 $$ The correct option is A. $\boxed{\text{A. } \sqrt{5} + 2}$ 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. $$ |z| = \sqrt{4 \times 2} $$ $$ |z| = \sqrt{4} \times \sqrt{2} $$ $$ |z| = 2\sqrt{2} $$ The modulus of $z$ is $\boxed{2\sqrt{2}}$.