Chapter 1.6, Problem 1ST

a.

To determine

Toexplain:

The property that simplifies the given simplification.

Answer to Problem 1ST

The property that simplifies the given simplification is commutative property.

Explanation of Solution

Given information:

simplifications:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

The equation in the first step is the form $\left(a+b\right)+c=a+\left(b+c\right)$ . Thus, the property used in simplification is the commutative property.

Hence, the property that simplifies the given simplification is commutative property.

b.

To determine

To explain:

The property that simplifies the given simplification.

Answer to Problem 1ST

The property that simplifies the given simplification is associative property.

Explanation of Solution

Given information:

simplifications:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

With the help of given expression first rewrite the second step, the given expression is as follows:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

Rewrite the second step:

  $\frac{1}{2}\left[-x+\left(x+2\right)\right]=\frac{1}{2}\left[\left(-x+x\right)+2\right]$

As know that this of the form $\left(a+b\right)+c=a+\left(b+c\right)$ , which is the associative property. The property used in simplification is the associative property.

Hence, the property that simplifies the given simplification is associative property.

c.

To determine

To explain:

The property that simplifies the given simplification.

Answer to Problem 1ST

The property that simplifies the given simplification is oppositesproperty.

Explanation of Solution

Given information:

simplifications:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

With the help of given expression copy the third step, the given expression is as follows:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

Copy the step:

  $\frac{1}{2}\left[-x+\left(x+2\right)\right]=\frac{1}{2}\left[0+2\right]$

As know that this of the form $a+\left(-a\right)=0$ , therefore it is inferring that the property of opposites has been used.

Hence, the property that simplifies the given simplification is opposites property.

d.

To determine

To explain:

The property that simplifies the given simplification.

Answer to Problem 1ST

The property that simplifies the given simplification is identifyproperty.

Explanation of Solution

Given information:

simplifications:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

With the help of given expression rewrite the fourth step, the given expression is as follows:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

Rewrite the fourth step:

  $\frac{1}{2}\left[0+2\right]=\frac{1}{2}\cdot 2$

As know that this of the form $a+0=a$ , which is the identify the property.

Hence, the property that simplifies the given simplification is identify property.

e.

To determine

To explain:

The property that simplifies the given simplification.

Answer to Problem 1ST

The property that simplifies the given simplification is reciprocals property.

Explanation of Solution

Given information:

simplifications:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

With the help of given expression copy the fifth step, the given expression is as follows:

  $\begin{array}{l}\frac{1}{2}\left[\left(x+2\right)+\left(-x\right)\right]=\frac{1}{2}\left[-x+\left(x+2\right)\right]\\ =\frac{1}{2}\left[\left(-x+x\right)+2\right]\\ =\frac{1}{2}\left(0+2\right)\\ =\frac{1}{2}\cdot 2\\ =1\end{array}$

Copy the fifth step:

  $\frac{1}{2}\cdot 2=1$

As know that this of the form $a.\frac{1}{a}=1$ , which is the reciprocals the property.

Hence, the property that simplifies the given simplification is reciprocals property.