Question

Solve each absolute value equation or inequality and choose the correct answer from the choices provided

1. [X]+5=18
A.5 or-5

B.13 or-13

C. 18 or -18

D. 23 or-23


2. [x+3]<5

A. -8
B. -2
C. 3
D.-3

3. [-3n] -2=4

A. 2 or -2

B. 3 or -3

C. 4 or -4

D. 6 or -6

Answer 1

Answers:

1)
`|x|+5=18`

If we want to solve equations with absolute values we must know we have to find the solution for both positive and negative values. This is because positive and negative values have a positive absolute value.

In a mathematical form this is:

For any positive number
`a`, the solution to
`|x|=a` is:

`|x|=a` or
`|x|=-a`

In this case we have to clear
`|x|` first:

`|x|=18-5`

`|x|=13`

This means
`x=13` or
`x=-13`

Therefore, the answer is B

2)
`|x+3|<5`

In the case of inequalities we have the following statement:

For any positive value of
`a`:

`|x|<a` is equivalent to
`-a<x<a`

`|x|>a` is equivalent to
`x<-a` or
`x>a`

Where
`x` may be a normal variable or an algebraic expression, as the expression in this exercise.

According to the explained above:

`|x+3|<5` is equivalent to
`-5< x+3<5`

This means we have to solve the inequality for both cases.

Case 1:

`x+3<5`

`x<5-3`

`x<2`

Case 2:

`x+3>-5`

`x>-5-3`

`x>-8`

Then,
`x<2` or
`x>-8`

3)
`|-3n|-2=4`

`|-3n|=4+2`

`|-3n|=6`

This means
`|-3n|=6` or
`|-3n|=-6`

Case 1:

`-3n=6`

`n=-(6)/(3)`

`n=-2`

Case 2:

`-3n=-6`

`n=(-6)/(-3)`

`n=2`

Then, the answer is A:
`n=-2` or
`n=2`