Solving Absolute Value Inequalities
To solve the inequality \( |u - 7| > 29 \), we need to split it into two separate inequalities because the absolute value expression can represent a positive or negative result:
1. If the expression inside the absolute value, \( u - 7 \), is positive, then \( u - 7 > 29 \). To solve for \( u \), add 7 to both sides of the inequality:
\( u - 7 + 7 > 29 + 7 \)
\( u > 36 \)
2. If the expression inside the absolute value, \( u - 7 \), is negative, then \( -(u - 7) > 29 \). This is equivalent to \( 7 - u > 29 \). To solve for \( u \), first subtract 7 from both sides:
\( 7 - u - 7 > 29 - 7 \)
\( -u > 22 \)
Next, multiply both sides by -1 to isolate \( u \), and remember to reverse the inequality sign when multiplying or dividing by a negative number:
\( -u(-1) < 22(-1) \)
\( u < -22 \)
Combine both solutions to express the full solution to the original inequality:
\( u > 36 \) or \( u < -22 \)
Hence, \( u \) is any number greater than 36 or any number less than -22.
