Find all real solutions. (Enter your answers as comma-separated lists. If there is no real solution, enter NO REALSOLUTION.) [tex]4|x-5|=10[/tex]
Answer (2)
Divide both sides of the equation by 4: ∣ x − 5∣ = 2 5 .
Split the absolute value equation into two cases: x − 5 = 2 5 and x − 5 = − 2 5 .
Solve for x in both cases: x = 5 + 2 5 = 2 15 and x = 5 − 2 5 = 2 5 .
The solutions are 2 5 , 2 15 .
Explanation
Problem Analysis We are given the equation 4∣ x − 5∣ = 10 and asked to find all real solutions for x . This involves solving an absolute value equation.
Isolating the Absolute Value First, we divide both sides of the equation by 4 to isolate the absolute value term: 4 4∣ x − 5∣ = 4 10 ∣ x − 5∣ = 2 5
Splitting into Cases Now, we consider the two cases for the absolute value:
Case 1: x − 5 = 2 5
Case 2: x − 5 = − 2 5
Solving Case 1 Solving for x in Case 1: x = 5 + 2 5 x = 2 10 + 2 5 x = 2 15 = 7.5
Solving Case 2 Solving for x in Case 2: x = 5 − 2 5 x = 2 10 − 2 5 x = 2 5 = 2.5
Final Answer Therefore, the solutions are x = 2 15 and x = 2 5 .
Examples
Absolute value equations are useful in various real-world scenarios, such as determining the tolerance in manufacturing processes. For example, if a machine is designed to produce parts that are 5 cm in length, but a tolerance of 0.5 cm is allowed, the actual length, x , must satisfy the equation ∣ x − 5∣ ≤ 0.5 . Solving this inequality helps determine the acceptable range of lengths for the manufactured parts.
To solve the equation 4∣ x − 5∣ = 10 , we first isolate the absolute value to get ∣ x − 5∣ = 2 5 . This leads to two cases which yield the solutions x = 7.5 and x = 2.5 . Hence, the real solutions are x = 2 15 , 2 5 .
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