Solving Absolute Value Inequalities
To solve the inequality \( |6 - w| < 59 \), we approach it by considering the definition of the absolute value.
The absolute value \( |a| \) of a number is its distance from zero on the number line, regardless of direction.
This means \( |6 - w| < 59 \) tells us that the expression \( 6 - w \) must be less than 59 units away from 0 on the number line.
We must consider two cases due to the absolute value:
1. \( 6 - w \) is positive or zero:
\[ 6 - w < 59 \]
\[ -w < 59 - 6 \]
\[ -w < 53 \]
\[ w > -53 \]
(When we divide or multiply by a negative number, we reverse the inequality sign).
2. \( 6 - w \) is negative:
\[ -(6 - w) < 59 \]
\[ w - 6 < 59 \]
\[ w < 59 + 6 \]
\[ w < 65 \]
Therefore, combining both cases, the solution to the inequality is:
\[ -53 < w < 65 \]
This tells us w must be greater than -53 and less than 65.
