Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the value of each expression. $$-|-9-(-6)|-(-12)$$

Step-by-Step Solution

Answer

-3

Step 1: Simplify Inside the Absolute Value

Begin by simplifying the expression within the absolute value brackets. That is, solve \(9 - (-6)\), which equals \(9 + 6 = 15\). So we now have \(-|15| - (-12)\).

Step 2: Apply Absolute Value

Next, apply the absolute value operation. The absolute value of 15 is 15. Hence, the expression becomes \(-15 - (-12)\).

Step 3: Simplify the Expression

Solve \(-15 - (-12)\), which equals to \(-15 + 12 = -3\).

Key Concepts

Simplifying Expressions

Simplifying expressions is about breaking down complex expressions into more manageable parts. The goal is to make it easier to find the actual value of the expression. Think of it as cleaning up a messy equation. You want the simplest form.
When you simplify expressions involving absolute values, begin by handling any operations inside those absolute value bars first. Things like subtraction or addition should be calculated before you take the absolute value. This keeps your work organized and straightforward.
Start by solving any operations in the innermost parts of the expression. Then, move outward, simplifying step by step. This is the path to clarity. Consider it a process much like following a recipe, where each step builds on what’s been done already.

Integer Operations

Operations on integers include addition, subtraction, multiplication, and division. Integers are whole numbers, which can be positive, negative, or zero. Being comfortable with these operations is critical.
To perform addition and subtraction with integers, it's handy to remember the rules involving signs:

• Adding two positive integers or two negative integers gives a sum with the same sign as the integers being added.
• Adding a positive integer and a negative integer is like subtracting the smaller absolute value from the larger absolute value.
• When subtracting a number, you add its opposite. For example, \(-(-6)\) becomes \(+6\).

Using these rules on your given expression ensures you handle each integer operation correctly, resulting in an accurate simplification and final result.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. In simpler terms, they are equations or formulas with some placeholders that can take different values. In algebra, knowing how to simplify such expressions becomes a powerful skill.
Even though not directly dealing with variables in this exercise, the process of simplifying can be applied to more complex algebraic expressions in the same way. Look for opportunities to combine like terms, factor expressions, or clear fractions. You'll declutter the expression and reveal the true value hiding inside.
With practice, you'll see that these techniques can help solve not only straightforward numerical problems but also those involving variables and unknowns, making complex algebraic problems feel more approachable.