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Chapter 3: Problem 57

Apply the distributive property, then simplify. $$\frac{2}{3}(-3 x-6)$$

Short Answer

Expert verified
The expression simplifies to \(-2x - 4\).

Step by step solution

01

Identify the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). Here, we have a difference \(-3x - 6\). We will distribute \(\frac{2}{3}\) to both \(-3x\) and \(-6\).
02

Distribute to the First Term

Multiply \(\frac{2}{3}\) by \(-3x\). This is calculated as follows: \(\frac{2}{3} \times (-3x) = -2x\).
03

Distribute to the Second Term

Multiply \(\frac{2}{3}\) by \(-6\). This is calculated as follows: \(\frac{2}{3} \times (-6) = -4\).
04

Combine the Results

Write the expression as the sum of the results from steps 2 and 3. This results in \(-2x - 4\).

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They are like a language, communicating mathematical ideas through symbols and numbers. In our exercise, we have the expression \(\frac{2}{3}(-3x - 6)\).
  • Here, \(-3x - 6\) combines a variable term \(-3x\) and a constant term \(-6\).
  • The variable \(x\) can take different values. Consequently, the value of the entire expression changes with \(x\).
  • Algebraic expressions are simplified using rules like the distributive property and combining like terms.
Algebraic expressions allow us to solve problems where numbers alone are insufficient. They offer flexibility and dynamism, as they can represent a range of scenarios.
Making Simplification Simple
Simplification aims to make expressions easier to work with by reducing their complexity. For instance, reducing \(\frac{2}{3}(-3x - 6)\) results in \(-2x - 4\). Why simplify?
  • Focuses on reducing clutter and confusion by removing unnecessary elements.
  • Finds a concise way to express the same mathematical idea.
To simplify:
  • Identify and apply mathematical rules, like the distributive property, effectively.
  • Combine like terms whenever possible.
Simplifying isn't just about shorter expressions. It's about gaining clarity and ensuring calculations are manageable.
The Art of Multiplying Fractions
Multiplying fractions seems tricky, but with a clear method, it's straightforward. Here's how we tackled multiplying \(\frac{2}{3}\) by each term separately in the expression.
  • Multiply the numerators together (top numbers) and the denominators together (bottom numbers).
  • Simplify the resulting fraction if possible.
In our solution:
  • For \(\frac{2}{3} \times (-3x)\), multiply \(2\) by \(-3x\) and divide by \(3\), simplifying to \(-2x\).
  • For \(\frac{2}{3} \times (-6)\), multiply \(2\) by \(-6\) and divide by \(3\), resulting in \(-4\).
Remember: fraction multiplication doesn't need common denominators, and it's all about straight multiplication and simplifying the results.

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Most popular questions from this chapter

The following problems all involve the concept of borrowing. Subtract in case. \(8-1 \frac{3}{4}\)

Combine. \(3 \frac{2}{3}+4 \frac{1}{3}\)

Add. $$\frac{8}{4}+\frac{3}{4}$$

A dress manufacturer usually buys two rolls of cloth, one of \(32 \frac{1}{2}\) yards and the other of \(25 \frac{1}{3}\) yards, to fill his weekly orders. If his orders double one week, how much of the cloth should he order? (Give the total yardage.)

Subtract \(\frac{7}{8}\) from the product of 2 and \(3 \frac{1}{2}\).
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