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Chapter 2: Problem 33

Simplify the expression. $$6 t \div\left(-\frac{1}{2}\right)$$

Short Answer

Expert verified
-12t

Step by step solution

01

Identify the reciprocal of the denominator

The reciprocal of \(-\frac{1}{2}\) is \(-2\). Remember that the reciprocal of a fraction is obtained by swapping the numerator and denominator.
02

Replace division operation with multiplication

Replace the division operation in the expression \(6t \div -\frac{1}{2}\) with multiplication, this gives us \(6t \times -2\).
03

Perform the multiplication operation

Multiply 6 and -2 and keep the \(t\) attached. The result is \(-12t\)

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Key Concepts

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

Reciprocal
Understanding the concept of a reciprocal is crucial in mathematics, especially when simplifying expressions. The reciprocal of a number is simply one divided by that number. For a fraction, this means swapping the numerator and the denominator.
For instance, the reciprocal of \(-\frac{1}{2}\) is \(-2\). This is because when you flip \(-\frac{1}{2}\), it becomes \(-2\).
If you multiply a number by its reciprocal, the result is always 1. This property is useful in various mathematical operations, such as division, where finding reciprocals can simplify complex calculations.
Division to Multiplication
Sometimes, division in expressions can be simplified by converting it into a multiplication problem. This technique involves using the reciprocal of the divisor.
For the expression \(6t \div -\frac{1}{2}\), we replace division by multiplying by the reciprocal. Thus, division by \(-\frac{1}{2}\) becomes multiplication by \(-2\).
This switch is beneficial because multiplication is generally more straightforward and less error-prone, especially when dealing with complex fractions or variables.
Multiplication of Integers
Once division is converted into multiplication, the next step is to carry out the multiplication of integers. In the expression \(6t \times -2\), simply multiply the integers 6 and -2.
Performing this multiplication, we get \(-12\), and since we initially had the variable \(t\) attached, the simplified expression becomes \(-12t\).
Remember, when multiplying integers, combining a positive and a negative number always results in a negative product. Attention to these signs ensures accuracy in your final answer.

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Most popular questions from this chapter

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -s(7+s) $$

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MULTI-STEP PROBLEM A customer of your flower shop wants to send flowers to 23 people. Each person will receive an \(\$ 11.99\) "sunshine basket" or a \(\$ 16.99\) "meadow bouquet." a. Let \(s\) represent the number of people who will receive a sunshine basket. Which function can you use to find \(C\), the total cost of sending flowers to all 23 people, depending on how many of each arrangement is sent? (A) \(C=16.99(23-s)+11.99 s\) (B) \(C=11.99 s+16.99(23)\) b. If 8 people receive a sunshine basket, what is the total cost of the flowers? c. If 13 people receive a meadow bouquet, what is the total cost of the flowers? d. CRITICAL THINKING If your customer can spend only \(\$ 300\), what is the greatest number of people that can receive a meadow bouquet?

BUYING JEANS You have 58 dollar, and you want to buy a pair of jeans and a \(\$ 20\) T-shirt. There is a \(6 \%\) sales tax. If \(x\) represents the cost of the jeans, then the following inequality is a model that shows how much you can spend on the jeans. $$ x+20+0.06(x+20) \leq 58 $$ If the jeans cost \(\$ 35,\) can you buy both the T-shirt and the jeans?

INVESTING MONEY You receive \(\$ 5000\). You decide to invest the money in a one-year bond paying \(2 \%\) interest and in a one-year certificate of deposit paying \(6 \%\) interest. \- Let \(m\) represent the amount of money invested in the one-year bond. Write a function that represents the total amount of money \(T\) that you have after one year. Simplify the function.
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