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Chapter 7: Problem 3

Multiply \(-4(2 x-6)\). (a) \(-8 x+24\) (b) \(-8 x-24\) (c) \(-6 x+10\) \((d)-8 x-6\)

Short Answer

Expert verified
The expression simplifies to -8x + 24, so the correct answer is (a).

Step by step solution

01

Understand the expression

The expression you need to multiply is -4(2x - 6). This involves distributing the -4 across the terms inside the parentheses.
02

Distribute the first term

Multiply -4 by the first term inside the parentheses, 2x. Recall that to distribute means to multiply each term inside the parenthesis by the term outside. So, (-4) 3 (2x) equals -8x.
03

Distribute the second term

Next, multiply -4 by the second term inside the parentheses, which is -6. Remember that multiplying two negative numbers results in a positive number. Therefore, (-4) 3 (-6) equals +24.
04

Combine the results

Combine the products from Steps 2 and 3. This gives the final expression -8x + 24. This is the simplified form of the expression.
05

Verify the correct option

Compare the simplified expression -8x + 24 to the given options. Option (a) matches -8x + 24 exactly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Numbers
When working with negative numbers, it's important to understand their properties. Negative numbers are numbers less than zero and are often used to express a decrease or loss.
In multiplication, remember these key points:
  • Multiplying a negative number by a positive number results in a negative number. For example, \(-4 imes 2 = -8\).

  • Multiplying two negative numbers yields a positive result. This is because the negatives cancel each other out. So, \(-4 imes -6 = 24\).

  • When dealing with more than two factors and an odd number of negative factors, the product is negative. If there is an even number of negative factors, it鈥檚 positive.
Understanding these properties will greatly help in performing operations involving negative numbers, especially in multiplying algebraic expressions.
Multiplying Algebraic Expressions
When multiplying algebraic expressions, the distributive property is a handy tool. It tells us how to distribute multiplication over addition or subtraction within parentheses.
For the expression \(-4(2x-6)\), apply the distributive property as follows:
  • Multiply the outer term \(-4\) by each term inside the parenthesis, which is \(2x\) and \(-6\).

  • Starting with \(2x\), multiply to get \(-4 \times 2x = -8x\).

  • Next, multiply \(-4\) by \(-6\) to get \(-4 \times -6 = 24\).
By performing these steps, you distribute one expression across the terms of another. Always ensure you correctly handle the signs during multiplication.
Simplifying Expressions
Simplifying expressions involves reducing them into a more concise form without changing their meaning. It's like tidying up your work!
Here鈥檚 how to simplify the expression \(-4(2x - 6)\):
  • After distributing, the expression becomes \(-8x + 24\).

  • The goal is to ensure all terms are combined as much as possible and that no further simplification is possible. In this case, \(-8x\) and \(+24\) are already fully simplified.

  • Finally, compare your simplified version with any given options to ensure accuracy. This helps confirm that the simplified expression \(-8x + 24\) matches an available choice, verifying its correctness.
Practicing simplification aids in solving algebraic problems with greater ease and accuracy.

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