Problem 114 - Intermediate Algebra

Question

a. \(9 x-2(-3 x+4)\) b. \(9 x-2(-3 x)+4\)

Step-by-Step Solution

Answer

(a) \(15x - 8\), (b) \(15x + 4\).

Step 1: Identifying the Expression

First, let's identify the expressions we need to examine: (a) \(9x - 2(-3x + 4)\) and (b) \(9x - 2(-3x) + 4\). Each expression contains terms that involve multiplication and distribution, which we will tackle.

Step 2: Simplifying Expression (a)

For expression (a) \(9x - 2(-3x + 4)\), we distribute the \(-2\) across the terms in the parentheses:\(-2(-3x + 4) = (-2) \times (-3x) + (-2) \times 4 = 6x - 8\).Next, substitute back:\(9x + 6x - 8\).Combine like terms:\(15x - 8\).

Step 3: Simplifying Expression (b)

For expression (b) \(9x - 2(-3x) + 4\), we distribute \(-2\) to \(-3x\):\(-2 \times (-3x) = 6x\).Thus, the expression becomes:\(9x + 6x + 4\).Combine like terms:\(15x + 4\).

Key Concepts

Distributive Property in Algebra

In algebra, the distributive property is a fundamental concept that helps us to simplify expressions, especially when dealing with parentheses. This property allows us to multiply a single term by each term inside the parentheses. This comes in handy when you are simplifying expressions and equations.
When using the distributive property, it looks like this: for any numbers or variables, if you have something like \(a(b + c)\), it can be rewritten as \(a \cdot b + a \cdot c\). This means you multiply the term outside the parentheses by each term inside the parentheses and sum up the results.

• In expression (a) \(9x - 2(-3x + 4)\), we distribute \(-2\) over \(-3x + 4\). So, the process is similar: \(-2 \cdot (-3x) + (-2 \cdot 4) = 6x - 8\).
• In expression (b) \(9x - 2(-3x) + 4\), \(-2\) distributes only to \(-3x\), leading to \(6x\).

By applying the distributive property, simplifying algebraic expressions becomes more manageable and it paves the way for further operations like combining like terms.

Combining Like Terms

Once the distributive property has been applied, the next step often involves combining like terms, another critical aspect of simplifying algebraic expressions.
Like terms are terms that have the same variables raised to the same power. For example, \(9x\) and \(6x\) are like terms because they both contain the variable \(x\) to the first power. This makes them easily addable or subtractable from one another.
In simplifying expressions, you gather these like terms to create a less complicated expression.

• For expression (a) that simplifies to \(9x + 6x - 8\), after using the distributive property, the like terms \(9x\) and \(6x\) are combined for a total of \(15x\).
• Similarly, for expression (b), the terms \(9x\) and \(6x\) result in \(15x\). The standalone constant, \(4\) in expression (b), stays the same and is added at the end.

This process is crucial not just for simplifying but also for making algebraic expressions solvable or more readable.

Simplification of Expressions

Simplification in algebra refers to the process of altering an expression into its simplest form where no further operations (like factoring, distributing, or combining) can be done. This involves several steps like applying the distributive property, and combining like terms as previously discussed.
The goal is to make expressions as neat and compact as possible, simplifying problem-solving in algebra, and making it easier to compare or solve equations.
For our original exercises, the simplifications achieved are:

• In expression (a), the final simplified form is \(15x - 8\).
• In expression (b), it is \(15x + 4\).

There are no more like terms to combine and no further distribution is possible in these simplified expressions. Each expression has been reduced to its most concise form, ready for immediate use in any calculations or further algebraic operations. Ultimately, simplification helps make algebra more understandable and manageable.