Question

Indicate whether each of the following equations is sure to have a solution set of all real numbers. Explain your

answers for each.
a. 3(x + 1) = 3x + 1
b. x + 2 = 2 + x
c. 4x(x + 1) = 4x + 4x²
d. 3x(4x)(2x) = 72x³

Answer 1

a)
`0 = -2` is not a valid equality. So a) will not have a solution set of all real numbers.

b)
`0 = 0` is a valid equality for all real numbers. So b) will have a solution set of all real numbers.

c)
`0 = 0` is a valid equality for all real numbers. So c) will have a solution set of all real numbers.

d)
`24x^(3) = 72x^(3)` is only valid for
`x = 0`. So d) will not have a solution set of all real numbers.

Explanation:

a)

`3(x+1) = 3x + 1`

`3x + 3 = 3x + 1`

`3x - 3x = 1 - 3`

`0 = -2`

`0 = -2` is not a valid equality. So a) will not have a solution set of all real numbers.

b)

`x + 2 = 2 + x`

`x - x = 2 - 2`

`0 = 0`

`0 = 0` is a valid equality for all real numbers. So b) will have a solution set of all real numbers.

c)

`4x(x+1) = 4x + 4x^(2)`

`4x^(2) + 4x = 4x + 4x^(2)`

`4x^(2) - 4x^(2) = 4x - 4x`

`0 = 0`

`0 = 0` is a valid equality for all real numbers. So c) will have a solution set of all real numbers.

d)

`3x(4x)(2x) = 72x^(3)`

`24x^(3) = 72x^(3)`

This is only valid for
`x = 0`. So d) will not have a solution set of all real numbers.