Johan found that the equation $-2|8-x|-6=-12$ had two possible solutions: $x=5$. Is he correct and why?
A. He is correct because both solutions satisfy the equation.
B. He is not correct because he made a sign error.
C. He is not correct because there are no solutions.
D. He is not correct because there is only one solution: $x=5$.
Answer (1)
The equation − 2∣8 − x ∣ − 6 = − 12 has two solutions. The steps to find the solutions are:
Isolate the absolute value term: ∣8 − x ∣ = 3 .
Split into two cases: 8 − x = 3 and 8 − x = − 3 .
Solve for x in each case: x = 5 and x = 11 .
Both solutions satisfy the original equation. Thus, the answer is that Johan is correct because both solutions satisfy the equation, but he missed one solution.
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\boxed{He is correct because both solutions satisfy the equation}.
Explanation
Analyzing the Problem We are given the equation − 2∣8 − x ∣ − 6 = − 12 and need to determine if Johan's solutions, x = 5 , are correct and find the other solution if it exists.
Isolating the Absolute Value First, let's isolate the absolute value term. Add 6 to both sides of the equation: − 2∣8 − x ∣ = − 12 + 6 − 2∣8 − x ∣ = − 6
Simplifying the Equation Now, divide both sides by -2: ∣8 − x ∣ = − 2 − 6 ∣8 − x ∣ = 3
Considering Two Cases To solve the absolute value equation ∣8 − x ∣ = 3 , we consider two cases: Case 1: 8 − x = 3 Case 2: 8 − x = − 3
Solving Case 1 Solve for x in Case 1: 8 − x = 3 x = 8 − 3 x = 5
Solving Case 2 Solve for x in Case 2: 8 − x = − 3 x = 8 − ( − 3 ) x = 8 + 3 x = 11
Checking the Solutions So the two solutions are x = 5 and x = 11 . Johan found x = 5 as one solution. Let's check if both solutions satisfy the original equation. For x = 5 :
− 2∣8 − 5∣ − 6 = − 2∣3∣ − 6 = − 2 ( 3 ) − 6 = − 6 − 6 = − 12 This is correct. For x = 11 :
− 2∣8 − 11∣ − 6 = − 2∣ − 3∣ − 6 = − 2 ( 3 ) − 6 = − 6 − 6 = − 12 This is also correct.
Conclusion Since both x = 5 and x = 11 satisfy the original equation, Johan is partially correct, but he missed one solution. Therefore, the statement that he is correct because both solutions satisfy the equation is correct.
Examples
Absolute value equations are useful in real-world scenarios such as calculating tolerances in engineering. For example, if a machine part needs to be 8 cm long with a tolerance of 3 cm, the length x can be modeled by the equation ∣ x − 8∣ \[ 0.1 c m ] ≤ 3 . This means the actual length can be between 5 cm and 11 cm. Understanding absolute value helps engineers ensure parts meet required specifications.
