Chapter 1.9, Problem 53MR

(a)

To determine

Determine whether the sentence $x+x=x$ is sometimes, always or never true.

Answer to Problem 53MR

Never

Explanation of Solution

Given:

The sentence $x+x=x$

Concept Used:

Addition or Subtraction Property of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}\text{+}\text{c}\text{=}\text{b}\text{+}\text{c}$ or If $\text{a}\text{=}\text{b}$ , then $\text{a}-\text{c}\text{=}\text{b}-\text{c}$

The property that states that if you add or subtract the same number to both sides of an equation, the sides remain equal (i.e., the equation continues to be true.)

Calculation:

In the sentence $x+x=x$, the left side $x+x=2x$ but the right side is $x$.

  $2x\ne x$

Thus, the sentence $x+x=x$ is never true.

(b)

To determine

Determine whether the sentence $x+0=x$ is sometimes, always or never true.

Answer to Problem 53MR

Always true.

Explanation of Solution

Given:

The sentence: $x+0=x$

Concept Used:

Addition or Subtraction Property of Equality:

Identity Property of Zero: The Identity Property of Zero states that the sum of zero and any number is that number.

Calculation:

The Identity Property of Zero states that the sum of zero and any number is that number

According to this property addition of zero and any number is that number.

  $x+0=x$ is always true.

Thus, the sentence $x+0=x$ is always true.