Problem 47 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 2(x-2)+4 $$

Step-by-Step Solution

Answer

For the given expression, after simplifying, the result is \(2x\).

Step 1: Distributive Property

To get rid of the parentheses, the term \(2\) needs to be distributed to every term inside the parentheses. Hence, the expression becomes: \[ 2*x - 2*2 + 4 \]

Step 2: Simplify the Multiplication

Multiply the numbers accordingly. The expression now looks as:\[ 2x - 4 + 4 \]

Step 3: Combine Like Terms

Finally, combine similar terms. There are no similar terms with \(2x\), but \(-4\) and \(+4\) can be summed. This results in:\[ 2x + 0 \]

Key Concepts

Distributive Property

Understanding the distributive property is essential for simplifying algebraic expressions. It allows you to multiply a single term by each term within a set of parentheses. Think of distribution in math as sharing or spreading out the values evenly.

For example, if you have the expression \(2(x - 2) + 4\), the distributive property would be used to multiply \(2\) by every term inside the parentheses: \(x\) and \( -2\). The expression then becomes \(2x - 4 + 4\) after you've 'distributed' the \(2\).

Combining Like Terms

After using the distributive property, we often need to simplify the expression further by combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, \(2x\) and \(3x\) are like terms because both contain the variable \(x\) to the first power.

In the expression \(2x - 4 + 4\), there are no like terms to combine with \(2x\), but we do have two constants: \( -4\) and \( +4\). Since these are like terms, you can add them together, which results in \(0\). Thus, the expression simplifies to \(2x + 0\), which is finally just \(2x\), as the addition of zero does not change the value.

Symbols of Grouping

Symbols of grouping like parentheses \( ( ) \), brackets \[ [ ] \], or braces \( \{ \} \) are used in algebra to indicate which operations should be performed first according to the order of operations. In the given expression \(2(x - 2) + 4\), parentheses are used to group \(x - 2\) together, signifying that you must first address the subtraction before applying the distributive property.

Once the operation inside the parentheses is carried out or distributed over, the symbols of grouping have served their purpose and can be removed. In subsequent steps, we look for and combine like terms to simplify the expression further, ultimately eliminating the need for grouping symbols in our solution.