Given the function [tex]f(x)=4|x-5|+3[/tex], for what values of [tex]x[/tex] is [tex]f(x)=15[/tex]?
A. [tex]x=2, x=8[/tex]
B. [tex]x=1.5, x=8[/tex]
C. [tex]x=2, x=7.5[/tex]
D. [tex]x=0.5, x=7.5[/tex]
Answer (2)
The values of x that satisfy the equation f(x) = 15 for the function f(x) = 4|x - 5| + 3 are x = 2 and x = 8. Therefore, the correct answer is option A: x = 2, x = 8.
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Set up the equation 4∣ x − 5∣ + 3 = 15 .
Isolate the absolute value: ∣ x − 5∣ = 3 .
Split into two cases: x − 5 = 3 and x − 5 = − 3 .
Solve for x in each case: x = 8 and x = 2 . The solution is x = 2 , x = 8 .
Explanation
Understanding the Problem We are given the function f ( x ) = 4∣ x − 5∣ + 3 , and we want to find the values of x for which f ( x ) = 15 . This involves solving an absolute value equation.
Setting up the Equation First, we set f ( x ) equal to 15: 4∣ x − 5∣ + 3 = 15
Isolating the Absolute Value Next, we isolate the absolute value term. Subtract 3 from both sides of the equation: 4∣ x − 5∣ = 15 − 3
4∣ x − 5∣ = 12
Simplifying the Equation Now, divide both sides by 4: ∣ x − 5∣ = 4 12
∣ x − 5∣ = 3
Considering Two Cases To solve the absolute value equation ∣ x − 5∣ = 3 , we consider two cases:
Case 1: x − 5 = 3 Case 2: x − 5 = − 3
Solving for x Solve for x in each case:
Case 1: x − 5 = 3 . Add 5 to both sides: x = 3 + 5 = 8
Case 2: x − 5 = − 3 . Add 5 to both sides: x = − 3 + 5 = 2
Final Answer Therefore, the values of x for which f ( x ) = 15 are x = 2 and x = 8 .
Examples
Absolute value equations are useful in many real-world scenarios, such as determining tolerances in manufacturing. For example, if a machine is designed to produce parts that are 5 cm long, but a tolerance of 0.3 cm is allowed, the actual length x of the part must satisfy the equation ∣ x − 5∣ ≤ 0.3 . This ensures that the part is within acceptable limits. Similarly, in physics, absolute values are used to calculate distances, speeds, and other quantities that are always non-negative.
