Solve for x: |x - 2| < 3

Analysis

The question asks us to solve an inequality involving an absolute value: $|x - 2| < 3$.

Concept

The absolute value of a number represents its distance from zero on a number line. When we have an inequality of the form $|f(x)| < a$ (where $a > 0$), it means the expression inside the absolute value must be between $-a$ and $a$.

Mathematically, this translates to: $-a < f(x) < a$

Solution

Given the inequality: $|x - 2| < 3$

Step 1: Rewrite the inequality without the absolute value bars. Based on the principle above, we set the expression between $-3$ and $3$: $-3 < x - 2 < 3$

Step 2: Isolate $x$. To get $x$ by itself in the middle, we need to add $2$ to all three parts of the inequality: $-3 + 2 < x - 2 + 2 < 3 + 2$

Step 3: Simplify. $-1 < x < 5$

This means $x$ must be greater than $-1$ and less than $5$.

Conclusion

Comparing our result to the given options: A. $x < 5$ B. $-2 < x < 3$ C. $-1 < x < 5$ D. $x < 1$

The correct option is C.

Exam Tip

Whenever you see an absolute value inequality with a "less than" sign ($<$), remember the acronym "LESS THAN = BETWEEN". This reminds you that the solution will be a range trapped between two values. If the sign were "greater than" ($>$), the solution would be "OR" (pointing outwards away from the center).