Question

solve the equation below
3|x - 4| + 1 = 19
x =

Explanation:

Step1: Subtract 1 from both sides

To isolate the absolute - value term, we first subtract 1 from both sides of the equation \(3|x - 4|+1 = 19\).

$$3|x - 4|+1-1=19 - 1$$

$$3|x - 4|=18$$

Step2: Divide both sides by 3

Next, we divide both sides of the equation \(3|x - 4| = 18\) by 3 to get the absolute - value expression alone.

$$\frac{3|x - 4|}{3}=\frac{18}{3}$$

$$|x - 4| = 6$$

Step3: Solve the absolute - value equation

The absolute - value equation \(|x - 4| = 6\) means that \(x-4 = 6\) or \(x - 4=-6\) (because if \(|a|=b\) where \(b\geq0\), then \(a = b\) or \(a=-b\)).

Case 1: \(x - 4=6\)

Add 4 to both sides of the equation \(x - 4 = 6\):

$$x-4 + 4=6 + 4$$

$$x = 10$$

Case 2: \(x - 4=-6\)

Add 4 to both sides of the equation \(x - 4=-6\):

$$x-4 + 4=-6 + 4$$

$$x=-2$$

Answer:

\(x = 10\) or \(x=-2\)