Question

solve for u.
|u - 7| - 4 ≤ 8
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3. use integers fractions, or improper fractions in simplest form.

Explanation:

Step1: Isolate the absolute value

Add 4 to both sides of the inequality \(|u - 7| - 4 \leq 8\) to get \(|u - 7| \leq 8 + 4\).
\(|u - 7| \leq 12\)

Step2: Solve the absolute value inequality

The absolute value inequality \(|a| \leq b\) (where \(b\geq0\)) is equivalent to \(-b \leq a \leq b\). So for \(|u - 7| \leq 12\), we have \(-12 \leq u - 7 \leq 12\).

Step3: Solve for u

Add 7 to all parts of the compound inequality: \(-12 + 7 \leq u - 7 + 7 \leq 12 + 7\).
Simplify each part: \(-5 \leq u \leq 19\)

Answer:

\(-5 \leq u \leq 19\)