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Chapter 1: Problem 87

Which property of real numbers justifies each statement? $$-5(x+4)=-5 x+(-5)(4)$$

Short Answer

Expert verified
The property of real numbers that justifies the given statement is the distributive property

Step by step solution

01

Identify the property shown

Look at the situation where -5 is being multiplied by each item within the parentheses. This is showing that -5 times the quantity of something can also be looked at as -5 times each individual part. So, -5 times x and -5 times 4 are being calculated separately.
02

Verifying the property

To verify the property, calculate -5 times x and then -5 times 4 separately. Next, compute -5 times the quantity of x + 4. If the sum of -5 times x and -5 times 4 equals to the product of -5 and the quantity of x + 4, the distributive property is verified.
03

Conclude the property

After comparing the results, it is obvious that the product of -5 times the quantity of x + 4 equals to the sum of -5 times x and -5 times 4. Therefore, the property of real numbers that justifies the given statement is the distributive property.

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Key Concepts

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

Real Numbers
Real numbers include all the numbers you encounter daily. They encompass whole numbers, fractions, and irrational numbers like 蟺. Essentially, everything on the number line falls into this category.
Real numbers are crucial in algebra because they allow us to work with a wide range of values. This flexibility is vital when solving equations and performing operations like addition, subtraction, multiplication, and division.
  • Whole numbers: 0, 1, 2, 3, ...
  • Fractions: \( \frac{1}{2}, \frac{2}{3} \, \rm{etc.} \)
  • Irrational numbers: numbers that cannot be expressed as a simple fraction, like 鈭�2 or 蟺
This concept helps us understand various mathematical properties, like the distributive property, which simplifies algebraic operations.
Multiplication
Multiplication is one of the essential arithmetic operations, finding how many times a number is added to itself. When you multiply real numbers, you find the product of these numbers.
This operation is associative and commutative, meaning you can multiply in any order and group without changing the result.
For example, \[(3 \times 4) \times 5 = 3 \times (4 \times 5)\]Understanding multiplication within algebraic expressions is crucial to applying properties like the distributive property. Here, multiplication interacts with addition or subtraction within an expression to simplify calculations.
Algebraic Expressions
Algebraic expressions contain numbers, variables, and arithmetic operations like addition or multiplication. They represent real-world situations mathematically with the help of variables that stand in for unknown values.
For example, in the expression \( 2x + 3y \), \( x \) and \( y \) are variables.
Expressions are simplified using properties like the distributive property. For instance, the expression \( -5(x + 4) = -5x + (-5 \times 4) \) illustrates how we distribute multiplication over addition.
  • Terms: Parts of the expression separated by + or - signs.
  • Coefficients: Numbers multiplying the variables.
  • Constants: Numbers not attached to variables, like "4" in the example above.
This organization and simplification lead to solving complex equations efficiently.

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Most popular questions from this chapter

A woman earns \(\$ 48,712.32\) in taxable income. She must pay \(15 \%\) tax on the first 823,000 and \(28 \%\) on the rest. In addition, she must pay a Social Security tax of \(15.4 \%\) on the total amount. How much tax will she need to pay?

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) The quotient obtained when the product of \(x\) and \(y\) is divided by \(z\)

Write each sentence as a mathematical expression. \(\sqrt{2}\) is approximately equal to 1.41

Use the given property to rewrite the expression in a different form. SEE EXAMPLE \(6 .\) (OBJECTIVE 5 ) \((x+y) z ;\) commutative property of addition.

A plumber cuts a pipe that is 12 feet long into \(x\) equal pieces. Find an expression for the length of each piece.
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