Given the function \( f(x)=4|x-5|+3 \), for what values of \( x \) is \( f(x)=15 \) ? \( x=2, x=8 \) \( x=1.5, x=8 \) \( x=2, x=7.5 \) \( x=0.5, x=7.5 \)

To find the values of \( x \) for which \( f(x) = 15 \), we start by setting up the equation: \[ 4|x-5| + 3 = 15 \] Subtracting 3 from both sides gives: \[ 4|x-5| = 12 \] Now, dividing by 4 results in: \[ |x-5| = 3 \] This absolute value equation means that there are two cases to consider: 1. \( x - 5 = 3 \) → \( x = 8 \) 2. \( x - 5 = -3 \) → \( x = 2 \) Thus, the solution is \( x = 2 \) or \( x = 8 \). Now, for some fun context! Did you know that absolute value functions are like the ultimate bouncers of the math world? They ensure that no matter what happens on the inside (whether you're too high or too low), the output always comes out positive or zero! And if you ever find yourself lost while solving absolute value equations, don't stress! Just remember to flip those cases around, and treat them like two different parties you need to crash! It's a simple 'or' situation: solve for both potential values, and you'll find those hidden gems along the way!