Problem 60 - Algebra 1: Concepts and Skills

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7) $$ -2(x+8)+36 $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x+20\).

Step 1: Distribution

Distribute the -2 across (x+8). -2 times x is -2x and -2 times 8 is -16.

Step 2: Combine Like Terms

Add the constant terms together. In this case, it is -16 and 36, which gives 20.

Step 3: Write Final Simplified Expression

The final simplified expression after combining terms is -2x+20.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by each term within a set of parentheses. It is especially useful when simplifying algebraic expressions that include parentheses. In the expression \(-2(x+8) + 36\), let's see how the distributive property applies.

• First, identify the number outside the parentheses, which in this example is -2.
• Distribute the -2 to each of the terms inside the parentheses: the x and the 8.
• When you multiply -2 by x, you get -2x.
• Then, multiply -2 by 8 to get -16.

By distributing -2 across \((x + 8)\), the expression transforms into \(-2x - 16\). Thus, the distributive property helps break down complex expressions into simpler parts by distributing a factor across terms inside the parentheses.

Combining Like Terms

After using the distributive property, the next step is combining like terms. Like terms are terms in an expression that have the same variable raised to the same power. Combining them helps to simplify expressions further.In our example, \(-2x - 16 + 36\), we identify:

• Numbers without a variable as like terms. Here, these are -16 and 36.

To combine them, simply add these constants together: -16 plus 36 gives you 20. This step consolidates these separate constant terms into a single number, allowing the expression to reduce to: \(-2x + 20\).This simplification ensures the expression is more concise and clearer, which can be quite helpful in solving algebraic equations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition or subtraction). These expressions can be simplified using properties such as the distributive property and combining like terms.Let's break down the expression \(-2(x + 8) + 36\) to understand this concept:

• It contains a variable term \(x\), which is a letter representing a number.
• There are numerical constants, such as -2, 8, and 36, which are numbers on their own.

Simplifying this expression involves handling each component systematically, using algebraic rules. First, apply the distributive property to eliminate parentheses, then combine like terms to make the expression shorter and more manageable.Understanding algebraic expressions and how to simplify them is crucial in algebra, as it lays the groundwork for solving equations and understanding functions.