Find the value of $x,$ if \[|x-20| + |x-18| = |2x-36|.\]
We recognize that $|2x-36| = 2|x-18|,$ so we get \[|x-20| = |x-18|.\]This means that, on the number line, $x$ is equidistant from $20$ and $18.$ Therefore $x$ must lie halfway between $20$ and $18,$ so \[x = \frac{20+18}{2} = \boxed{19}.\]
19
Level 2
Intermediate Algebra
Problem: Find the value of $x,$ if \[|x-20| + |x-18| = |2x-36|.\]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation |x-20| + |x-18| = |2x-36|, we first recognize that |2x-36| can be simplified to |2(x-18)|, which is 2|x-18|. This allows us to rewrite the equation as |x-20| + |x-18| = 2|x-18|. 

Next, we can rearrange the equation to isolate the absolute values: |x-20| = 2|x-18| - |x-18|, which simplifies to |x-20| = |x-18|. 

Now, we need to consider the cases for the absolute values. The critical points are x = 20 and x = 18. We will analyze the intervals defined by these points: 

1. Case 1: x < 18
   Here, |x-20| = 20 - x and |x-18| = 18 - x. The equation becomes 20 - x = 18 - x, which simplifies to 20 = 18, a contradiction. 

2. Case 2: 18 ≤ x < 20
   In this interval, |x-20| = 20 - x and |x-18| = x - 18. The equation becomes 20 - x = x - 18. Solving this gives 20 + 18 = 2x, or 38 = 2x, leading to x = 19.

3. Case 3: x ≥ 20
   In this case, |x-20| = x - 20 and |x-18| = x - 18. The equation becomes x - 20 = x - 18, which simplifies to -20 = -18, another contradiction.

The only valid solution occurs in Case 2, where we found x = 19. Thus, the final answer is 19.<|END_OF_LATENT|> The final answer is $19$.