1.9 Properties of real numbers  (Page 5/10)

Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in [link] would look like this:

Simplify: $8\left(\frac{3}{8}x+\frac{1}{4}\right).$

Solution

Distribute.
Multiply.

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Using the distributive property as shown in [link] will be very useful when we solve money applications in later chapters.

Simplify: $100\left(0.3+0.25q\right).$

Solution

Distribute.
Multiply.

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When we distribute a negative number, we need to be extra careful to get the signs correct!

Simplify: $\mathrm{-2}\left(4y+1\right).$

Solution

Distribute.
Multiply.

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Simplify: $\mathrm{-11}\left(4-3a\right).$

Solution

Distribute.
Multiply.
Simplify.

Notice that you could also write the result as $33a-44.$ Do you know why?

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[link] will show how to use the distributive property to find the opposite of an expression.

There will be times when we’ll need to use the distributive property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

All the properties of real numbers we have used in this chapter are summarized in [link] .

Key concepts

• Commutative Property of
  • Addition: If $a,b$ are real numbers, then $a+b=b+a.$
  • Multiplication: If $a,b$ are real numbers, then $a·b=b·a.$ When adding or multiplying, changing the order gives the same result.
• Associative Property of
  • Addition: If $a,b,c$ are real numbers, then $\left(a+b\right)+c=a+\left(b+c\right).$
  • Multiplication: If $a,b,c$ are real numbers, then $(a·b)·c=a·(b·c).$
    When adding or multiplying, changing the grouping gives the same result.
• Distributive Property: If $a,b,c$ are real numbers, then
  • $a\left(b+c\right)=ab+ac$
  • $(b+c)a=ba+ca$
  • $a\left(b-c\right)=ab-ac$
  • $\left(b-c\right)a=ba-ca$
• Identity Property
  • of Addition: For any real number $a\text{:}a+0=a0+a=a$
    0 is the additive identity
  • of Multiplication: For any real number $a\text{:}a·1=a1·a=a$
    $1$ is the multiplicative identity
• Inverse Property
  • of Addition: For any real number $a,a+\left(\text{−}a\right)=0.$ A number and its opposite add to zero. $\text{−}a$ is the additive inverse of $a.$
  • of Multiplication: For any real number $a,(a\ne 0)a·\frac{1}{a}=1.$ A number and its reciprocal multiply to one. $\frac{1}{a}$ is the multiplicative inverse of $a.$
• Properties of Zero
  • For any real number $a,$
    $a·0=00·a=0$ – The product of any real number and 0 is 0.
  • $\frac{0}{a}=0$ for $a\ne 0$ – Zero divided by any real number except zero is zero.
  • $\frac{a}{0}$ is undefined – Division by zero is undefined.