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Absolute value is a fundamental concept in mathematics. It's the distance a number is from zero on the number line, regardless of direction. This means it is always non-negative. For example:

• \[\begin{equation} |3| = 3 \end{equation}\]
• \[\begin{equation} |-3| = 3 \end{equation}\]

Even if the number is negative, its absolute value turns it into a positive number.

In our exercise, we have the absolute values of several terms:\[\begin{equation} |-5| = 5 \end{equation}\], \[\begin{equation} |1| = 1 \end{equation}\], and \[\begin{equation} |-6| = 6 \end{equation}\].
Notice how the absolute value 'flips' negative numbers to positive but leaves positive numbers as they are.

Simplifying expressions is an essential skill in algebra. It involves combining like terms, simplifying inside mathematical operations, and more. For this exercise, start with the given expression: \[\begin{equation} |1 + (-6)| + |1| + |-6| \end{equation}\].

First, calculate the expression inside the absolute value: \[\begin{equation} 1 + (-6) = -5 \end{equation}\].

Simplifying before taking the absolute value makes the math easier.
After simplifying the expressions, apply the absolute value to \[\begin{equation} -5\end{equation}\], \[\begin{equation} 1 \end{equation}\], and \[\begin{equation} -6 \end{equation}\].
This gives us \[\begin{equation} |-5| = 5, \end{equation}\] \[\begin{equation} |1| = 1, \end{equation}\] and \[\begin{equation} |-6| = 6 \end{equation}\].

Adding integers is a core arithmetic operation. It can involve positive and negative numbers. For this exercise, you need to add the absolute values of \[\begin{equation} -5\end{equation}\], \[\begin{equation} 1\end{equation}\], and \[\begin{equation} -6\end{equation}\].

The absolute values we found were \[\begin{equation} 5, 1, \end{equation}\] and \[\begin{equation} 6 \end{equation}\].

Now, simply add them together:

• \[\begin{equation} 5 + 1 = 6 \end{equation}\]
• \[\begin{equation} 6 + 6 = 12 \end{equation}\]

The final value is \[\begin{equation} 12 \end{equation}\]. This solution shows how absolute values and integer addition come together.