Solving Equations with Roots and Absolute Values

Question

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?

Answer 1

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

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

$$15-2\sqrt{x-18}=0.5x-13$$

$$28-0.5x=2\sqrt{x-18}$$

$$56-x=4\sqrt{x-18}$$

Squaring both sides,

$$(56-x)^2=16(x-18)$$

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.

Answer 2

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.