Problem 57 - Elementary Algebra

Question

State the restrictions and then simplify. $$ x+66+x $$

Step-by-Step Solution

Answer

The expression simplifies to \(2x + 66\) with no restrictions.

Step 1: Identify Restrictions

The expression given is a simple algebraic expression. Typically, we look for restrictions when there are denominators, square roots over variables, or logarithms involved, because these can cause undefined expressions (like division by zero or negative numbers under a square root). In this case, there are no such operations, so there are no restrictions for the variable \(x\).

Step 2: Combine Like Terms

Simplify the expression by combining like terms. In this case, both terms \(x\) are like terms. The expression can be rewritten as \(2x + 66\).

Step 3: Write the Simplified Expression

Now that all like terms have been combined, the simplified expression is \(2x + 66\).

Key Concepts

Algebraic Restrictions

In algebra, when we talk about restrictions, we typically refer to limitations or conditions that a variable must satisfy within an expression. Such algebraic restrictions often arise in expressions involving division, square roots, or logarithms.
These operations pose specific challenges as they can render an expression undefined under certain conditions. For example:

• Division by zero is undefined, so expressions like \( \frac{1}{x} \) dictate that \( x eq 0 \).
• Negative numbers under a square root or logarithmic function also lead to undefined expressions in the real number system.

In our exercise, the expression \( x + 66 + x \) does not involve any such operations. Therefore, there are no algebraic restrictions. This highlights the simplicity and freedom of working with linear expressions that do not limit the values that the variables can take.

Like Terms

The concept of like terms is fundamental in simplifying algebraic expressions. Like terms are terms in an expression that have the same variable raised to the same power. This property allows them to be combined through addition or subtraction because they represent similar kinds of quantities.
Consider a simple expression like \( x + 66 + x \). Here:

• The two \( x \) terms are like terms because they both involve the variable \( x \) raised to the first power (\( x^1 \)).
• Numbers without variable parts like \( 66 \) are constants and separate from the like terms involving variables.

Identifying and combining like terms is a key step in the simplification process, allowing us to rewrite the expression as \( 2x + 66 \). This not only simplifies handling the expression but also prepares it for further algebraic operations.

Simplification Process

Simplifying algebraic expressions is a core skill in algebra that involves cleaning up expressions by combining like terms, reducing complexity, and making them more manageable.
The process generally involves:

• Identifying and combining like terms, as demonstrated in our exercise with the expression \( x + 66 + x \), ultimately simplifying to \( 2x + 66 \).
• Performing operations such as addition or subtraction on coefficients; here, the coefficients of the term \( x \) (which were both 1, as in 1x) were combined.
• Rewriting the expression to make it cleaner and more concise.

Through simplification, expressions become easier to interpret and solve, especially when solving equations or analyzing functions. In educational exercises like these, understanding the simplification process empowers students to handle more complex algebraic expressions with confidence.