Q.$x$ and $b$ are real numbers. If $b > 0$ and $|x| > b$, then
(A) $x \in (-b, \infty)$
(B) $x \in [-\infty, b)$
(C) $x \in (-b, b)$
(D) $x \in (-\infty, -b) \cup (b, \infty)$

The key idea here is solving Linear Inequalities involving Absolute Values.

For any real number $x$ and a positive real number $b$, the inequality $|x| > b$ means that the distance of $x$ from zero is greater than $b$. This condition is satisfied if $x$ is either greater than $b$ or less than $-b$.

1. The definition of $|x| > b$ (for $b > 0$) translates to two separate inequalities: $x > b$ or $x < -b$.
2. The solution to $x > b$ is the interval $(b, \infty)$.
3. The solution to $x < -b$ is the interval $(-\infty, -b)$. …

When we encounter an inequality involving an absolute value, like $|x| > b$, the most effective way to understand it is by thinking about distance on the number line. The absolute value of a number, $|x|$, represents its distance from zero.

So, the inequality $|x| > b$ means "the distance of $x$ from zero is strictly greater than $b$."

Since $b$ is given as a positive number ($b > 0$), we are looking for all real numbers $x$ that are more than $b$ units away from $0$.

Let's visualize this:

On a number line, the points that are exactly $b$ units away from $0$ are $b$ and $-b$.

If $x$ must be further than $b$ units away from $0$, then $x$ cannot be between $-b$ and $b$ (inclusive). Instead, $x$ must lie to the left of $-b$ or to the right of $b$.

This leads us to two separate conditions for $x$:

1. $x$ is to the right of $b$, meaning $x > b$.
2. $x$ is to the left of $-b$, meaning $x < -b$.

Since $x$ can satisfy either of these conditions to fulfill $|x| > b$, we combine them using the logical "OR". In set notation, "OR" corresponds to the union ($\cup$) of the solution sets.

Here is the step-by-step solution:

1. Interpret the absolute value inequality:

   The given inequality is $|x| > b$.

   As discussed, $|x|$ represents the distance of $x$ from $0$ on the number line.

   The condition $|x| > b$ means that the distance of $x$ from $0$ must be strictly greater than $b$.

2. Formulate equivalent linear inequalities:

   For $x$ to be more than $b$ units away from $0$, $x$ must satisfy one of two conditions:

   • $x$ is greater than $b$ (i.e., $x > b$).
   • $x$ is less than $-b$ (i.e., $x < -b$). These two conditions are connected by "OR".

   For any real number $x$ and any positive real number $b$, the inequality $|x| > b$ is equivalent to $x < -b$ or $x > b$.

3. Express the solutions in interval notation:

   • The solution for $x > b$ is the open interval $(b, \infty)$. This includes all numbers strictly greater than $b$.
   • The solution for $x < -b$ is the open interval $(-\infty, -b)$. This includes all numbers strictly less than $-b$. …