Problem 56 - Elementary Algebra

Question

State the restrictions and then simplify. $$ 3 x-22-3 x $$

Step-by-Step Solution

Answer

The simplified expression is \(-22\) with no restrictions on \(x\).

Step 1: Identify the expression components

The given expression is \(3x - 22 - 3x\). It consists of two terms involving \(x\) and a constant: \(3x\), \(-22\), and \(-3x\).

Step 2: Combine like terms

Combine the \(x\) terms: \(3x\) and \(-3x\). These terms cancel each other out, leaving: \(-22\).

Step 3: Simplify the expression

After combining like terms, the simplified expression is \(-22\). There are no further terms to combine or simplify.

Step 4: State restrictions

The expression \(3x - 22 - 3x\) has no restrictions on \(x\) because no variable appears in the final simplified expression. Therefore, there are no domain restrictions. In the context of rational expressions or denominators, since there's no division by zero, \(x\) can take any real value.

Key Concepts

like terms

In algebra, like terms are terms that contain the same variables raised to the same power. The only difference between like terms is their coefficient or numerical part. Recognizing like terms is crucial for simplifying algebraic expressions.
For instance, in the expression given as the problem, you see terms like:

• \(3x\)
• \(-3x\)

These are like terms because they both involve the variable \(x\) raised to the same power (which is 1 in this case). When you combine these like terms, you focus only on adding or subtracting their coefficients:

• Since \(3x - 3x = 0\), these terms effectively eliminate each other.

Understanding this concept helps to simplify complex expressions by reducing them to their simplest form through the combination of like terms.

algebraic expression

An algebraic expression is a mathematical phrase involving numbers, variables, and operators (like addition and subtraction). These expressions represent values and can be simplified or evaluated for different values of variables.
The given problem starts with the algebraic expression:

• \(3x - 22 - 3x\)

This expression has two parts involving the variable \(x\) and a constant number,

• The variable terms are \(3x\) and \(-3x\). These are terms associated with the variable \(x\).
• The constant term is \(-22\), which does not change with respect to \(x\).

Breaking down an algebraic expression into its components makes it easier to work with, as you can identify like terms for simplification or plug-in values for evaluation.

simplification process

The simplification process in algebra refers to making an expression more concise and readable by combining like terms and performing basic arithmetic operations. Simplifying algebraic expressions is a fundamental skill for solving equations. Let's break down the simplification process step by step using the example provided:

• First, you identify and group like terms. In \(3x - 22 - 3x\), \(3x\) and \(-3x\) are like terms.
• Next, you combine these like terms: \(3x - 3x = 0\). Here, the \(x\) terms completely cancel each other out.
• The remaining constant, \(-22\), is the simplified form of the expression.

After simplification, the expression reduces to a much simpler form, \(-22\). It’s important to recognize that simplifying an expression doesn't change its value; it only presents it in a clearer form. In this case, there are no restrictions on \(x\) since there is no division or other operations that limit the domain.