Solve $|x+5|=5$ for $x$.
A. $x=-10,10$
B. $x=0$
C. $x=-10,0$
D. $x=-10$
E. all real numbers
F. no solution
Answer (2)
Split the absolute value equation into two cases: x + 5 = 5 and x + 5 = − 5 .
Solve the first case, x + 5 = 5 , by subtracting 5 from both sides to get x = 0 .
Solve the second case, x + 5 = − 5 , by subtracting 5 from both sides to get x = − 10 .
The solutions are x = 0 and x = − 10 , so the answer is x = − 10 , 0 .
Explanation
Understanding the Problem We are given the absolute value equation ∣ x + 5∣ = 5 and asked to solve for x . Absolute value equations can be solved by considering two cases.
Case 1: x+5 = 5 Case 1: The expression inside the absolute value is equal to 5. This gives us the equation x + 5 = 5 . To solve for x , we subtract 5 from both sides of the equation: x + 5 − 5 = 5 − 5 x = 0
Case 2: x+5 = -5 Case 2: The expression inside the absolute value is equal to -5. This gives us the equation x + 5 = − 5 . To solve for x , we subtract 5 from both sides of the equation: x + 5 − 5 = − 5 − 5 x = − 10
Final Answer Therefore, the solutions to the equation ∣ x + 5∣ = 5 are x = 0 and x = − 10 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, when manufacturing parts, there is often a tolerance for the dimensions of the parts. If a part is supposed to be 5 cm long, but it can be off by up to 0.1 cm, then the actual length x of the part must satisfy the equation ∣ x − 5∣ ≤ 0.1 . Solving this inequality would tell you the acceptable range of lengths for the part.
The solutions to the equation ∣ x + 5∣ = 5 are x = 0 and x = − 10 . Therefore, the answer is option C: x = − 10 , 0 .
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