Question

solve the inequality algebraically. write the solution in interval notation.
the absolute value of x-2/3 is less than or equal to 4

Answer 1

|x - 2/3| <= 4

if x - 2/3 >= 0 then |x - 2/3| = x - 2/3

In that case,

x - 2/3 <= 4
x <= 14/3 and x >= 2/3 (remember x - 2/3 >= 0)

so, x is in [2/3 , 14/3]

if x - 2/3 >= 0 then |x - 2/3| = -(x - 2/3)

In that case,

2/3 - x <= 4
x >= -10/3
But x - 2/3 < 0 means x < 2/3
So,
x is in [-10/3 , 2/3)

So, finally, x is in [-10/3 , 2/3)U[2/3 , 14/3]
or, x is in [-10/3 , 14/3]

This makes sense. Think of the graph of |x|. It is a V shape. If |x| < k, then you want the part of the V below the line y=k.

In this case, the V is shifted 2/3 to the right, and we want the part of the V below the line y=4