Question

Which of the following are solutions to [x+3|= 4x - 7? Check all that apply.

A. X = 1/2
B. x = -1/3
1 c. x= -11 /
D. x = 10

Answer 1

x =
`(10)/(3)`

Explanation:

Given

| x + 3 | = 4x - 7

The absolute value function always returns a positive value, however the expression inside can be positive or negative, thus

x + 3 = 4x - 7 OR - (x + 3) = 4x - 7

Solving the 2 equations

x + 3 = 4x - 7 ( subtract 4x from both sides )

- 3x + 3 = - 7 ( subtract 3 from both sides )

- 3x = - 10 ( divide both sides by - 3 )

x =
`(10)/(3)`

OR

- (x + 3) = 4x - 7, that is

- x - 3 = 4x - 7 ( subtract 4x from both sides )

- 5x - 3 = - 7 ( add 3 to both sides )

- 5x = - 4 ( divide both sides by - 5 )

x =
`\frac{4}5}`

------------------------------------------------------------

As a check

Substitute these values into the equation and if both sides are equal then they are the solutions.

x =
`(10)/(3)`

|
`(10)/(3)` + 3 | = |
`(19)/(3)` | =
`(19)/(3)` and 4(
`(10)/(3)` ) - 7 =
`(40)/(3)` -
`(21)/(3)` =
`(19)/(3)`

left side = right side

Thus x =
`(10)/(3)` is a solution

-----------------------------------------

x =
`(4)/(5)`

|
`(4)/(5)` + 3 | = |
`(19)/(5)` | =
`(19)/(5)` and 4(
`(4)/(5)` ) - 7 =
`(16)/(5)` -
`(35)/(5)` = -
`(19)/(5)`

left side ≠ right side

Thus x =
`(4)/(5)` is an extraneous solution