Problem 55 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ -2+[(8-11)-(-2-9)] $$

Step-by-Step Solution

Answer

The simplified expression is \(-16\).

Step 1: Simplify the Inner Expressions

Begin by simplifying the expressions inside the parentheses. We have two separate parts: \((8-11)\) and \((-2-9)\). Start with \(8-11\), which simplifies to \( -3\).

Step 2: Simplify the Second Inner Expression

Now simplify \((-2-9)\). Combine the numbers: \(-2 - 9 = -11\). So, the entire sub-expression is \(-(-11)\), which is \(+11\) because subtracting a negative is the same as adding a positive.

Step 3: Evaluate the Expression Inside the Brackets

Substitute the simplified parts into the bracket: \([(-3) - (11)]\). }Simplify this further by calculating \(-3 - 11\), which equals \(-14\).

Step 4: Combine with the Outer Expression

Now we have the expression \(-2 + [-14]\). Simplify by combining these values: \(-2 - 14 = -16\).

Key Concepts

Simplification

When simplifying arithmetic expressions, it's important to follow the order of operations carefully. Begin with any computations inside parentheses, then handle exponents, followed by multiplication and division, and finally addition and subtraction. This ensures that you're making calculations in the correct sequence to arrive at the accurate result. In the exercise, we start by addressing expressions within the parentheses,

• The expression within the parentheses 8-11 simplifies to -3 .
• Similarly, simplify -2-9 to -11 and transform it to +11 through negation.

Finally, replace these results back into the expression and follow computation until all operations are resolved. Remember to work systematically, step by step.

Arithmetic Expressions

Arithmetic expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. The key to tackling these expressions lies in understanding the proper order to execute operations.

• Operations within parentheses are completed first.
• Negative numbers require careful handling to ensure correct relationships are maintained.

When simplifying an arithmetic expression like the given one, follow a strict sequence, handling components methodologically. Inside the exercise, after dealing with parentheses, you trace through calculation steps such as

• Converting operations like -(-11) into +11 .

Understanding the flow of these operations will make arithmetic expressions clearer and more manageable.

Negative Numbers

Negative numbers can be a bit tricky, but with practice, they become easier to handle. They behave slightly differently than positive numbers, especially in operations such as addition and subtraction.

• Subtracting a negative is equivalent to adding its positive counterpart. For example, the operation -(-11) is actually +11 .
• Additionally, it's essential to note how negative values change the terms in an equation, as you might have observed when the second expression changed from -11 to +11 .

Each step involving negative numbers requires careful attention to maintain accuracy in the solution process. Fully grasping the concept of subtraction and negation ensures clarity and correctness in arithmetic expressions.