Given the function [tex]f(x)=6|x-2|+3[/tex], for what values of [tex]x[/tex] is [tex]f(x)=39[/tex]?

[tex]x=-8, x=4[/tex]
[tex]x=8, x=-4[/tex]
[tex]x=9, x=-4[/tex]
[tex]x=8, x=-10[/tex]

The values of x for which the function f ( x ) = 6∣ x − 2∣ + 3 equals 39 are x = 8 and x = − 4 .
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Answered by Anonymous | 2025-07-04

Set up the equation 6∣ x − 2∣ + 3 = 39 .
Isolate the absolute value term: ∣ x − 2∣ = 6 .
Split into two cases: x − 2 = 6 and x − 2 = − 6 .
Solve for x in each case to get x = 8 and x = − 4 . The final answer is x = 8 , x = − 4 ​ .

Explanation

Understanding the Problem We are given the function f ( x ) = 6∣ x − 2∣ + 3 and we want to find the values of x for which f ( x ) = 39 . This involves solving an absolute value equation.

Setting up the Equation First, we set f ( x ) equal to 39: 6∣ x − 2∣ + 3 = 39

Isolating the Absolute Value Term Next, we subtract 3 from both sides of the equation: 6∣ x − 2∣ = 39 − 3
6∣ x − 2∣ = 36

Simplifying the Equation Now, we divide both sides by 6: ∣ x − 2∣ = 6 36 ​
∣ x − 2∣ = 6

Considering Two Cases To solve the absolute value equation ∣ x − 2∣ = 6 , we consider two cases:


Case 1: x − 2 = 6
Case 2: x − 2 = − 6

Solving Case 1 For Case 1, we add 2 to both sides: x = 6 + 2
x = 8

Solving Case 2 For Case 2, we add 2 to both sides: x = − 6 + 2
x = − 4

Final Answer Therefore, the values of x for which f ( x ) = 39 are x = 8 and x = − 4 .


Examples
Absolute value equations are useful in many real-world scenarios, such as determining tolerances in manufacturing. For example, if a machine is supposed to cut a metal rod to a length of 10 cm, but there is a tolerance of 0.1 cm, the actual length x must satisfy the equation ∣ x − 10∣ ≤ 0.1 . This means the length can be between 9.9 cm and 10.1 cm. Similarly, absolute value equations can be used in navigation to calculate distances and in finance to model deviations from expected values.

Answered by GinnyAnswer | 2025-07-04