Question
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Answer
The solutions are \( u = -7 \) and \( u = 15 \).
Solution
Solve the equation \( 4|u-4|+10=54 \).
Solve the equation by following steps:
- step0: Solve for \(u\):
\(4\left|u-4\right|+10=54\)
- step1: Move the expression to the left side:
\(4\left|u-4\right|+10-54=0\)
- step2: Subtract the numbers:
\(4\left|u-4\right|-44=0\)
- step3: Separate into possible cases:
\(\begin{align}&4\left(u-4\right)-44=0,u-4\geq 0\\&4\left(-\left(u-4\right)\right)-44=0,u-4<0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&u=15,u\geq 4\\&u=-7,u<4\end{align}\)
- step5: Find the intersection:
\(\begin{align}&u=15\\&u=-7\end{align}\)
- step6: Rewrite:
\(u_{1}=-7,u_{2}=15\)
The solutions to the equation \(4|u-4|+10=54\) are \(u=-7\) and \(u=15\).
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Simplify this solution Bonus Knowledge
To solve for \( u \), we first isolate the absolute value term. Start with the equation: \[ 4|u-4| + 10 = 54 \] Subtract 10 from both sides: \[ 4|u-4| = 44 \] Now divide both sides by 4: \[ |u-4| = 11 \] This absolute value equation can be solved by considering the two cases: 1. \( u - 4 = 11 \) 2. \( u - 4 = -11 \) For the first case: \[ u - 4 = 11 \implies u = 15 \] For the second case: \[ u - 4 = -11 \implies u = -7 \] Thus, the solutions for \( u \) are: \[ u = 15 \quad \text{and} \quad u = -7 \]
