Given the function [tex]$f(x)=-5|x+1|+3$[/tex], for what values of [tex]$x$[/tex] is [tex]$f(x)=-12$[/tex] ?
A. [tex]$x=-2, x=-4$[/tex]
B. [tex]$x=-2, x=4$[/tex]
C. [tex]$x=2, x=-4$[/tex]
D. [tex]$x=2, x=4$[/tex]
Answer (1)
We are given the function f ( x ) = − 5∣ x + 1∣ + 3 and want to find the values of x such that f ( x ) = − 12 .
Substitute f ( x ) = − 12 into the equation: − 12 = − 5∣ x + 1∣ + 3 .
Isolate the absolute value term: ∣ x + 1∣ = 3 .
Solve for x + 1 = 3 and x + 1 = − 3 , which gives x = 2 and x = − 4 .
The solutions are x = 2 and x = − 4 , so the final answer is x = 2 , x = − 4 .
Explanation
Problem Setup We are given the function f ( x ) = − 5∣ x + 1∣ + 3 and we want to find the values of x for which f ( x ) = − 12 .
Substitution First, we substitute f ( x ) = − 12 into the equation: − 12 = − 5∣ x + 1∣ + 3
Isolating the Absolute Value Next, we isolate the absolute value term. Subtract 3 from both sides: − 12 − 3 = − 5∣ x + 1∣ − 15 = − 5∣ x + 1∣
Simplifying Now, divide both sides by -5: − 5 − 15 = ∣ x + 1∣ 3 = ∣ x + 1∣
Solving for x To solve the absolute value equation ∣ x + 1∣ = 3 , we consider two cases:
Case 1: x + 1 = 3 Subtract 1 from both sides: x = 3 − 1 x = 2
Case 2: x + 1 = − 3 Subtract 1 from both sides: x = − 3 − 1 x = − 4
Final Answer Therefore, the values of x for which f ( x ) = − 12 are x = 2 and x = − 4 .
Examples
Absolute value functions are used in various real-world applications, such as calculating distances or deviations from a target value. For example, in manufacturing, if a machine is set to produce parts of a certain length, the absolute value function can be used to determine how much each part deviates from the target length, regardless of whether the part is too long or too short. This helps in quality control and ensuring that the parts meet the required specifications. Similarly, in finance, absolute value can be used to measure the volatility of an investment, representing the magnitude of price changes without considering the direction.
