Problem 54
Question
Identify the algebraic expression that does not belong with the other three. Explain your reasoning. \(-6(x-2),\) \(x+12-7 x,\) \(-x-5 x+12,\) \(-6 x-12\)
Step-by-Step Solution
Verified Answer
The expression that does not belong is \(-6x - 12\) since all others simplify to \(-6x + 12\).
1Step 1: Simplify Each Expression
First, let's simplify each of the given expressions to identify their forms.1. The first expression is \(-6(x-2)\). Distribute \(-6\) across the terms inside the parentheses: \[-6x + 12\]2. The second expression is \(x + 12 - 7x\). Combine like terms:\[-6x + 12\]3. The third expression is \(-x - 5x + 12\). Combine like terms:\[-6x + 12\]4. The fourth expression is \(-6x - 12\). It is already simplified.
2Step 2: Compare Simplified Expressions
Now, compare the simplified forms of each expression:1. \(-6x + 12\) (from \(-6(x-2)\))2. \(-6x + 12\) (from \(x + 12 - 7x\))3. \(-6x + 12\) (from \(-x - 5x + 12\))4. \(-6x - 12\)The first three expressions simplify to \(-6x + 12\), while the fourth one is \(-6x - 12\).
Key Concepts
Simplifying ExpressionsLike TermsDistributive Property
Simplifying Expressions
Simplifying expressions is like tidying up a room. You want everything in its right place, orderly and clear! In algebra, it means taking a complicated or messy expression and making it as straightforward as possible.
To simplify an expression, perform operations and combine like terms if possible. It involves removing unnecessary components such as redundant parentheses or combining terms that can be easily added or subtracted together. The goal is to write the expression in its most reduced form, which makes further calculations much easier.
For example, when given the expression \(-6(x - 2)\), you apply the distributive property to eliminate the parentheses, leading to \(-6x + 12\). In this process, you're breaking down each term within the parentheses so the expression is more streamlined and straightforward to comprehend.
Simplifying expressions helps you see what the equation truly represents and is a critical skill for solving algebraic problems efficiently.
To simplify an expression, perform operations and combine like terms if possible. It involves removing unnecessary components such as redundant parentheses or combining terms that can be easily added or subtracted together. The goal is to write the expression in its most reduced form, which makes further calculations much easier.
For example, when given the expression \(-6(x - 2)\), you apply the distributive property to eliminate the parentheses, leading to \(-6x + 12\). In this process, you're breaking down each term within the parentheses so the expression is more streamlined and straightforward to comprehend.
Simplifying expressions helps you see what the equation truly represents and is a critical skill for solving algebraic problems efficiently.
Like Terms
In algebra, understanding like terms is essential. Like terms refer to terms in an algebraic expression that have the same variables raised to the same power. These terms can be combined, which is helpful in simplifying expressions.
For example, in the expression \(x + 12 - 7x\), both \(x\) and \(-7x\) are like terms because they share the same variable \(x\). You can combine them: \(x - 7x = -6x\).
When you combine like terms, you're essentially collecting all the similar items in one spot, making the calculation neater and manageable. This process helps prevent errors and reduces complexity in mathematical expressions.
For example, in the expression \(x + 12 - 7x\), both \(x\) and \(-7x\) are like terms because they share the same variable \(x\). You can combine them: \(x - 7x = -6x\).
When you combine like terms, you're essentially collecting all the similar items in one spot, making the calculation neater and manageable. This process helps prevent errors and reduces complexity in mathematical expressions.
- Identify the coefficients: these are the numbers in front of the variables.
- Combine the coefficients: add or subtract as necessary.
Distributive Property
The distributive property is a crucial concept that allows you to multiply a single term by each term inside a set of parentheses. It's like sharing a box of chocolates where everyone gets some of each flavor!
Mathematically, this property is expressed as \(a(b + c) = ab + ac\). In the expression \(-6(x - 2)\), applying the distributive property gives \(-6 \cdot x + (-6 \cdot -2) = -6x + 12\).
Using the distributive property helps in simplifying expressions and solving equations. It ensures everything inside the brackets is correctly multiplied by the term outside, providing a clear and correct outcome.
Mathematically, this property is expressed as \(a(b + c) = ab + ac\). In the expression \(-6(x - 2)\), applying the distributive property gives \(-6 \cdot x + (-6 \cdot -2) = -6x + 12\).
Using the distributive property helps in simplifying expressions and solving equations. It ensures everything inside the brackets is correctly multiplied by the term outside, providing a clear and correct outcome.
- Useful for eliminating parentheses.
- Ensures all terms are accounted for properly during multiplication.
- Essential for expanding expressions before further simplification.
Other exercises in this chapter
Problem 53
Solve each equation. Check your solution. $$29=n+4$$
View solution Problem 54
Find each product mentally. Example $$\begin{aligned}15 \cdot 12 &=15(10+2) \\\&=150+30 \text { or } 180\end{aligned}$$. $$7 \cdot 14$$
View solution Problem 54
Divide. $$50 \div(-2)$$
View solution Problem 54
Solve each equation. Check your solution. $$k-12=-40$$
View solution