7. What is the solution of $4|x-3|-8=8$?
A. $x=\frac{19}{4}$ or $x=7$
B. $x=-1$ or $x=7$
C. $x=-1$ or $x=\frac{19}{4}$
D. $x=\frac{3}{4}$ or $x=\frac{19}{4}$
Answer (1)
Isolate the absolute value term: ∣ x − 3∣ = 4 .
Consider the case x − 3 ≥ 0 : x − 3 = 4 , which gives x = 7 .
Consider the case x − 3 < 0 : 3 − x = 4 , which gives x = − 1 .
The solutions are x = − 1 or x = 7 .
Explanation
Problem Analysis We are given the equation 4∣ x − 3∣ − 8 = 8 . Our goal is to find the solution(s) for x .
Isolating the Absolute Value First, we isolate the absolute value term. Add 8 to both sides of the equation: 4∣ x − 3∣ − 8 + 8 = 8 + 8
4∣ x − 3∣ = 16
Simplifying the Equation Next, divide both sides by 4: 4 4∣ x − 3∣ = 4 16
∣ x − 3∣ = 4
Solving for x in Both Cases Now, we consider two cases for the absolute value:
Case 1: x − 3 ≥ 0 , which means x ≥ 3 . In this case, ∣ x − 3∣ = x − 3 . So, we have: x − 3 = 4
Add 3 to both sides: x = 4 + 3
x = 7
Since 7 ≥ 3 , this solution is valid.
Case 2: x − 3 < 0 , which means x < 3 . In this case, ∣ x − 3∣ = − ( x − 3 ) = 3 − x . So, we have: 3 − x = 4
Subtract 3 from both sides: − x = 4 − 3
− x = 1
Multiply both sides by -1: x = − 1
Since − 1 < 3 , this solution is also valid.
Final Solutions Therefore, the solutions are x = 7 and x = − 1 .
Examples
Absolute value equations are useful in various real-world scenarios, such as calculating the tolerance in manufacturing. For example, if a machine is designed to produce parts that are 3 cm in length, but a tolerance of 0.1 cm is allowed, the actual length x of the part must satisfy the equation ∣ x − 3∣ ≤ 0.1 . Solving this inequality helps determine the acceptable range of lengths for the manufactured parts.
