Solving Absolute Value Equation

Let's solve this absolute value equation step by step. We have the absolute value of x squared minus five x plus six, plus the absolute value of x squared minus four, which equals five times the absolute value of x minus two.

The first step is to factor each quadratic expression inside the absolute value bars. We look for two numbers that multiply to six and add to negative five for the first term.

Notice that the absolute value of x minus two appears in every term. We can distribute the absolute value across the products since the absolute value of a times b is equal to the absolute value of a times the absolute value of b.

Because x minus two is common to all terms, we can find one solution directly by setting it equal to zero.

If x is not equal to two, then the absolute value of x minus two is not zero, and we can safely divide the entire equation by it.

Now we need to solve this simplified equation. This type of equation is often solved by considering different intervals based on the critical points where the expressions inside the absolute values change sign.

However, let's use a geometric interpretation. The sum of the distances from x to three and from x to negative two is exactly five.