Solve each absolute value equation and graph the solution:

The correct answer is: x ≤ -1 and x ≥ 5

Hint:
|x| is known as the absolute value of x. It is the non-negative value of x irrespective of its sign. The value of absolute value of x is given by

First, we solve this inequality by considering the two cases. Then we plot the graph on the x- axis, or the real line R in such a way that the graph satisfies the value of x from both the cases.
Step by step solution:
The given inequality is
|3x - 6| ≥ 9
We use the definition of |x|, which is

For, 3x - 6 ≤ 0,
We have
|3x - 6 |= -(3x - 6) ≥ 9
Multiplying  on both sides, we have
3x - 6 ≤ -9
Adding 6 on both sides, we get
3x ≤ -9 + 6
Or
3x ≤ -3
Dividing by 3 throughout, we get
x ≤ -1
For, 3x - 6 ≥ 0,
We have
|3x - 6|= 3x - 6 ≥ 9
Adding 6 on both sides, we have
3x ≥ 9 + 6
Or
3x ≥ 15
Dividing by 3 throughout, we get
x ≥ 5
Combining the above two solutions, we get the solution
x ≤ -1 and x ≥ 5
We plot the above inequality on the real line.
-2       -1         0         1         2          3         4         5         7          8
The points -1 and 5 are included in the graph.
Note:
The given inequality contains only one variable. So, the graph is plotted on one dimension, which is the real line. Geometrically, the absolute value of a number may be considered as its distance from zero regardless of its direction. The symbol |.| is pronounced as ‘modulus’. We read |x| as ‘modulus of x’ or ‘mod x’.