Problem 78 - Elementary Algebra

Question

Simplify. $$ (x-4)-1-(x+1)-1 $$

Step-by-Step Solution

Answer

The expression simplifies to \(-6\).

Step 1: Simplify Inside Parentheses

Identify and simplify expressions inside the parentheses first. For the expression \((x-4)-(x+1)\), remove the parentheses: \(x - 4 - x - 1\).

Step 2: Combine Like Terms

Notice that the \(x\) terms \(x\) and \(-x\) cancel each other out. This leaves: \(-4 - 1\). Then subtract \(1\) again: \(-4 - 1 - 1\).

Step 3: Perform Arithmetic Operations

Calculate the remaining arithmetic: \(-4 - 1 - 1 = -6\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental technique in algebra that simplifies expressions by grouping similar terms together. Like terms are terms that have the exact same variable part. For example, in an expression like

• \(3x + 2x\)

both terms contain the variable \(x\). Thus, they can be combined to give \(5x\). This process helps in reducing the complexity of algebraic expressions.

In the example given where you simplify \((x - 4) - 1 - (x + 1) - 1\), combining like terms occurs after the parentheses are addressed. The terms \(x\) and \(-x\) cancel each other out. This makes the expression easier to work with because you are left with constant terms only, which can then be easily simplified.

Parentheses in Algebra

Parentheses play a crucial role in algebra, as they indicate which operations should be performed first in an expression. They act like a set of rules that organize and structure calculations to ensure that the correct result is achieved.

• When you encounter parentheses, your first step is to evaluate the expression within them.
• Once evaluated, you can then move on to address any others outside the parentheses.

In the expression \((x - 4) - 1 - (x + 1) - 1\), the parentheses help separate different terms and ensure that we deal with them systematically. By removing the parentheses and simplifying, you allow for a clearer expression that can then have like terms combined and calculations completed accurately.

Arithmetic Operations

Arithmetic operations involve the basic mathematical processes of addition, subtraction, multiplication, and division. These operations are used to perform calculations that simplify or solve algebraic expressions.

The key here is following the correct order of operations, often recalled by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our example, after simplifying the expression and combining like terms, the arithmetic operation remaining is subtraction.

• We perform the subtraction step-by-step: writing it as \[-4 - 1 - 1\]
• This simplifies the expression to \(-6\).

Mastering arithmetic operations within algebraic contexts is crucial, as it influences the accuracy and simplicity of the final result.