I'm preparing for the ACTM State contest, and I stumbled across this problem:

Which of the following is the sum of the tens digit and the units digit of the solution of the following equation?

$$15-2\sqrt{x-18} = 0.5 |x+14|-20$$

I know how to solve radical equations and absolute value equations, but I'm not familiar on how to solve an equation with both. Can anyone help me out?


we need $x \geq 18$, hence $x+14 \geq0$, hence the problem becomes





Squaring both sides,


Hence the problem has now become a quadractic equation. Note that after solving for the quadratic root, you should substitute the solution back to check whether they are indeed a solution.


One way to manage absolute values is to split your single equation into two different cases. For instance,

$$|x+4| = 12 \Rightarrow \\ x+4 = 12,~ x+4 = -12 \Rightarrow \\ x=8,~ x=-16$$

Since you have eliminated the absolute value, you can proceed normally in the two cases.


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