Q.Fill in the blank: If $|x + 2| > 5$, then $x$ ___ $-7$ or $x$ ___ $3$.

The core concept here is solving Linear Inequality Solutions involving absolute values.

An inequality of the form $|u| > a$ (where $a > 0$) implies that $u$ must be either greater than $a$ or less than $-a$. This is because the distance of $u$ from zero is greater than $a$.

Applying this to $|x+2| > 5$:

1. We split the absolute value inequality into two separate linear inequalities: $x+2 > 5$ or $x+2 < -5$.
2. Solve the first inequality: …

When we encounter an absolute value inequality like $|x+2| > 5$, the first step is to understand what the absolute value symbol means. The expression $|A|$ represents the distance of $A$ from zero on the number line.

So, $|x+2| > 5$ means that the quantity $(x+2)$ must be more than 5 units away from zero. This can happen in two distinct ways:

1. $(x+2)$ is greater than 5 (i.e., to the right of 5 on the number line).
2. $(x+2)$ is less than -5 (i.e., to the left of -5 on the number line).

These two conditions are mutually exclusive for any given $x$, but both satisfy the original inequality. Therefore, we connect them with the logical operator "or".

Let's apply this understanding to solve the given problem.

1. Identify the components of the inequality.

   In our inequality, $|x+2| > 5$, we can identify $A = x+2$ and $B = 5$.

2. Split the absolute value inequality into two linear inequalities.

   Based on the property of absolute value inequalities, $|x+2| > 5$ translates into two separate conditions:

   • $x+2 > 5$
   • or $x+2 < -5$

3. Solve the first linear inequality.

   We take the first condition, $x+2 > 5$, and solve for $x$:

   $x+2 > 5$

   Subtract 2 from both sides:

   $x > 5 - 2$

   $x > 3$

4. Solve the second linear inequality.

   Now, we take the second condition, $x+2 < -5$, and solve for $x$:

   $x+2 < -5$

   Subtract 2 from both sides:

   $x < -5 - 2$

   $x < -7$

5. Combine the solutions. …