Q.State whether the following statement is True or False: If $|x| > 5$, then $x \in (-\infty, -5) \cup [5, \infty)$.

The key idea here is understanding absolute value inequalities and their representation in interval notation.

The inequality $|x| > 5$ means that the distance of $x$ from zero on the number line is greater than 5 units.

This translates to two separate conditions: $x < -5$ or $x > 5$.

In interval notation, $x < -5$ is represented as $(-\infty, -5)$, and $x > 5$ is represented as $(5, \infty)$.

Combining these with the "or" condition gives $x \in (-\infty, -5) \cup (5, \infty)$. …

To determine if the statement is True or False, we need to understand what each part of the statement means and then compare them. The core concept here is the definition of absolute value and how it translates into inequalities, which can then be represented using interval notation.

The absolute value of a number $x$, denoted as $|x|$, represents its distance from zero on the number line, regardless of direction. For example, $|5| = 5$ and $|-5| = 5$.

1. Analyze the condition: $|x| > 5$

   This inequality means that the distance of $x$ from zero must be strictly greater than $5$.

   On a number line, this implies two separate regions:

   • $x$ is to the right of $5$, meaning $x > 5$.
   • $x$ is to the left of $-5$, meaning $x < -5$.

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

   Applying this formula, $|x| > 5$ means $x < -5$ or $x > 5$.

   In interval notation, this set of numbers is $(-\infty, -5) \cup (5, \infty)$.

   Notice that the numbers $-5$ and $5$ themselves are not included in this set because the inequality is strict ($>$).

2. Analyze the proposed conclusion: $x \in (-\infty, -5) \cup [5, \infty)$

   This interval notation describes a set of numbers where:

   • $x$ belongs to $(-\infty, -5)$, meaning $x < -5$.
   • OR $x$ belongs to $[5, \infty)$, meaning $x \ge 5$.

   This means that $x$ can be any number strictly less than $-5$, or any number greater than or equal to $5$.

3. Compare the two sets

   Let's compare the set of numbers defined by $|x| > 5$ with the set of numbers defined by $x \in (-\infty, -5) \cup [5, \infty)$.

   • From step 1, $|x| > 5$ corresponds to $x \in (-\infty, -5) \cup (5, \infty)$.
   • From step 2, the proposed conclusion is $x \in (-\infty, -5) \cup [5, \infty)$.

   The difference between these two sets lies in the inclusion of the number $5$.

   • The condition $|x| > 5$ does not include $x=5$. If $x=5$, then $|5|=5$, which is not strictly greater than $5$.
   • The interval $(-\infty, -5) \cup [5, \infty)$ does include $x=5$. If $x=5$, then $5 \ge 5$ is true, so $x=5$ is part of this interval. …