Olson Tran
06/17/2024 · Senior High School
Solve the absolute value equation or indicate that the equation has no solution. \( 3\left|5-\frac{7}{2} x\right|+12=39 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \{\} . (Use integers or fractions for any numbers in the expression. Use a comma to separate answe B. The solution set is the empty set.
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Step-by-step Solution
Solve the equation \( 3\left|5-\frac{7}{2} x\right|+12=39 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3\left|5-\frac{7}{2}x\right|+12=39\)
- step1: Move the expression to the left side:
\(3\left|5-\frac{7}{2}x\right|+12-39=0\)
- step2: Subtract the numbers:
\(3\left|5-\frac{7}{2}x\right|-27=0\)
- step3: Separate into possible cases:
\(\begin{align}&3\left(5-\frac{7}{2}x\right)-27=0,5-\frac{7}{2}x\geq 0\\&3\left(-\left(5-\frac{7}{2}x\right)\right)-27=0,5-\frac{7}{2}x<0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=-\frac{8}{7},x\leq \frac{10}{7}\\&x=4,x>\frac{10}{7}\end{align}\)
- step5: Find the intersection:
\(\begin{align}&x=-\frac{8}{7}\\&x=4\end{align}\)
- step6: Rewrite:
\(x_{1}=-\frac{8}{7},x_{2}=4\)
The solution to the absolute value equation \(3\left|5-\frac{7}{2}x\right|+12=39\) is \(x=-\frac{8}{7}\) or \(x=4\).
Therefore, the correct choice is:
A. The solution set is \(\{-\frac{8}{7}, 4\}\).
Quick Answer
The solution set is \(\{-\frac{8}{7}, 4\}\).
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