Question

What is

`- |x + 2| + 1 = 4x - 6`
​

Answer 1

x = 1

Explanation:

`|a|=\left\{\begin{array}{ccc}a&for&a\geq0\\-a&for&a<0\end{array}\right\\\\|x+2|=\left\{\begin{array}{ccc}x+2&for&x+2\geq0\to x\geq-2\\-(x+2)&for&x<-2\end{array}\right`

`(1)\ x<-2\\\\-\bigg(-(x+2)\bigg)+1=4x-6\\\\+(x+2)+1=4x-6\qquad\text{combine like terms}\\\\x+(2+1)=4x-6\\\\x+3=4x-6\qquad\text{subtract 3 from both sides}\\\\x+3-3=4x-6-3\\\\x=4x-9\qquad\text{subtract}\ 4x\ \text{from both sides}\\\\x-4x=4x-4x-9\\\\-3x=-9\qquad\text{divide both sides by (-3)}\\\\(-3x)/(-3)=(-9)/(-3)\\\\x=3\\otin(1)`

`(2)\ x\geq-2\\\\-(x+2)+1=4x-6\\\\-x-2+1=4x-6\\\\-x+(-2+1)=4x-6\\\\-x-1=4x-6\qquad\text{add 1 to both sides}\\\\-x-1+1=4x-6+1\\\\-x=4x-5\qquad\text{subtract}\ 4x\ \text{from both sides}\\\\-x-4x=4x-4x-5\\\\-5x=-5\qquad\text{divide both sides by (-5)}\\\\(-5x)/(-5)=(-5)/(-5)\\\\x=1\in(2)`

Answer 2

x = 1

Explanation:

Given

- | x + 2 | + 1 = 4x - 6 ( subtract 1 from both sides )

- | x + 2 | = 4x - 7 ( multiply both sides by - 1 )

| x + 2 | = - 4x + 7

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

x + 2 = - 4x + 7 ( add 4x to both sides )

5x + 2 = 7 ( subtract 2 from both sides )

5x = 5 ( divide both sides by 5 )

x = 1

OR

- (x + 2) = - 4x + 7, that is

- x - 2 = - 4x + 7 ( add 4x to both sides )

3x - 2 = 7 ( add 2 to both sides )

3x = 9 ( divide both sides by 3 )

x = 3

As a check substitute these values into both sides of the equation and if both sides are equal then they are the solutions.

x = 1 : - |1 + 2| + 1 = - | 3 | + 1 = - 3 + 1 = - 2

right side = 4(1) - 6 = 4 - 6 = - 2 ← True

x = 3 : - |3 + 2| + 1 = - |5| + 1 = - 5 + 1 = - 4

right side = 4(3) - 6 = 12 - 6 = 6 ← False

Thus x = 3 is an extraneous solution and

x = 1 is the solution