Question

Show the work to solve |5x – 2| greater than or equal to 8

Describe the graph of the solution in words

Answer 1

Keep the definition of the absolute value in mind:
\[|x| = x\;\text{if}\; x \le 0\;\;\;|x| = -x\;\text{if}\; x \ge 0\]. So we distinguish two cases to be able to continue our discussion:

1) 5x - 2 > 0 => |5x-2| = 5x-2
Out of 5x - 2 > 0 follows
5x > 2
x > 2/5
So by supposing that 5x-2>0 it follows that x is greater than 2/5.
Now we look at |5x - 2| = 5x - 2 >= 8.
5x - 2 >= 8 | +2
5x >= 10 | :2
x >= 2
So all x greater or equal to 2 satisfy our inequality. Now we look at the second case

2) 5x - 2 < 0 equivalent to x < 2/5. From that follows BY DEFINITION (see the above)
5x - 2 < 0 => |5x - 2| = -(5x -2) = -5x + 2 >= 8
-5x + 2 >= 8 | - 2
-5x >= 6 |:(-5) when dividing by a negative number, the inequality turns!
x <= -6/5

So, our solutions are all x smaller than or equal to -6/5 and all x greater than or equal to 2. So, on the real line It's just anything left of or equal to -6/5 and anything right of or equal to 2

Answer 2

thank you