Algebra Problem by Kevin Tran (2)

Let $f(x) = |x-8|+|4x-6|-|3x+4|-4$ . We need to find integer solution for $f(x) = 0$ . Then we have:

$f(x) = \begin{cases} -x+8 -4x+6-(-3x-4) - 4 = 14 - 2x & \text{for } x < - \frac 43 & \implies f(x) > 16\frac 23 & \color{#D61F06} \small \text{No solution} \\ -x+8 -4x+6-(3x+4) - 4 = 6 - 8x & \text{for } - \frac 43 \le x < \frac 32 & \implies f \left(\frac 34\right) = 0 & \color{#D61F06} \small \text{Not integer solution} \\ -x+8 +4x-6-(3x+4) - 4 = -6 & \text{for } \frac 32 \le x < 8 & \implies f(x) = -6 & \color{#D61F06} \small \text{No solution} \\ x-8 +4x-6-(3x+4) - 4 = 2x - 22 & \text{for } x \ge 8 & \implies f(11) = 0 & \color{#3D99F6} \small \text{Integer solution} \end{cases}$

Therefore, there is only $\boxed{1}$ integer solution, when $x = 11$ .