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When evaluating expressions, especially those involving absolute values, you begin by simplifying the expression inside any absolute value operators. This means performing any operations like addition, subtraction, multiplication, or division within the parentheses or absolute value bars first. Simplifying makes it easier to evaluate the entire expression accurately. In the given exercise, you start by calculating the expression inside the absolute value:

• Evaluate \(-7 - 19\)
• The result becomes \(-26\)

After simplifying, apply the absolute value, which we will explain shortly. Remember, the order of operations is essential when simplifying expressions, commonly abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that each part of the calculation is handled correctly.

Integer operations refer to calculations that involve whole numbers, both positive and negative. The four basic operations include addition, subtraction, multiplication, and division. For addition and subtraction:

• If both integers have the same sign, you add their absolute values and keep the sign.
• If they have different signs, subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value.

In the exercise, \(-7 - 19\), both integers have negative signs. When you add \(-7\) and \(-19\), you really add \(|7 + 19|\) = 26\ resulting in \(-26\), keeping the negative sign. Remember, integer operations are crucial in many mathematical expressions, so it's essential to understand how to handle different signs correctly.

A number line is a visual representation to display numbers in a linear format, where distances are uniform, and each point corresponds to a number. It is particularly useful when understanding the concept of absolute value. Absolute value refers to how far a number is from zero on a number line, regardless of direction. In our example, \(-26\) is 26 units away from zero on the number line. Thus, its absolute value is \(|-26| = 26\). Using a number line makes it easier to grasp concepts such as negative numbers and absolute values:

• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• Absolute value ignores the direction, focusing solely on distance from zero.

When dealing with absolute values and integer operations, visualizing them on a number line can help clarify any misunderstandings and lead to more accurate computations.