Problem 72 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (3 x+8)(-2) $$

Step-by-Step Solution

Answer

The expression (3x+8)(-2) rewritten without parenthesis using the distributive property is \(-6x - 16\).

Step 1: Identify the expression to distribute

Firstly, identify that -2 needs to be distributed to both the terms inside the parenthesis that is \(3x\) and \(8\).

Step 2: Apply distributive property

Next step is to apply the distributive property which means multiply -2 with \(3x\) and then multiply -2 with \(8\). So, the expression becomes \(-2*3x + (-2*8)\).

Step 3: Simplify the expression

-2 multiplied with \(3x\) gives \(-6x\) and -2 multiplied with 8 gives -16. So, the expression simplifies to \(-6x - 16\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra, as it allows you to rewrite mathematical expressions in their simplest form. This is done by combining like terms and applying mathematical properties, such as the distributive property. When simplifying expressions, you'll want to reduce the number of terms and eliminate any unnecessary parentheses.

• Identify the algebraic terms: Separate constants from variables, ensuring each term is distinct.
• Look for opportunities to use mathematical properties: Apply identities like commutative, associative, and distributive properties to break down expressions.
• Combine like terms: Like terms are terms that have the exact same variable parts (e.g., same variable and exponents). Add or subtract them accordingly.

When working with negative numbers while simplifying, ensure that you carefully follow the rules for adding, subtracting, multiplying, or dividing these numbers.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols like addition and multiplication. They do not have an equality sign, which distinguishes them from equations.

• Components of algebraic expressions: These include constants, coefficients, variables, and terms.
• Constants are fixed numbers, whereas coefficients are numbers that multiply a variable.
• Variables represent unknown values and are usually denoted by letters like x or y.

Understanding how to manipulate these expressions is important because it allows you to model real-world situations and solve problems. The distributive property is crucial in rearranging expressions, especially when parentheses are involved. It simplifies the expression, helping you avoid unnecessary complications.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They might seem a bit tricky at first, but they're essential in expressing concepts like debt, temperature below zero, or any deficit.

• Basic rules of operations with negatives:
  • Adding a negative is like subtracting a positive (e.g., 5 + (-3) = 5 - 3).
  • Subtracting a negative number is like adding its opposite positive number (e.g., 5 - (-3) = 5 + 3).
  • Multiplying or dividing two negative numbers gives a positive result, while one negative and one positive yields negative.
• Think of the number line: Negative numbers go to the left of zero, so they decrease in value compared to positive numbers.

When using the distributive property, dealing with negative numbers requires careful attention to the signs to avoid mistakes in the final expression. Remember, keeping track of negative signs is crucial for correct simplification of expressions.