solve the inequality algebraically. write the solution in interval notation.
the absolute value of x-2/3 is less than or equal to 4
|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
To solve the inequality |x - 2/3| ≤ 4 algebraically, we need to consider two separate cases: the case when the expression inside the absolute value is positive, and the case when it is negative.
Case 1: x - 2/3 ≥ 0
When x - 2/3 is positive or zero, the absolute value can be removed, giving us the following equation:
x - 2/3 ≤ 4
We can solve this equation algebraically:
x ≤ 4 + 2/3
x ≤ 14/3
Case 2: x - 2/3 < 0
When x - 2/3 is negative, the absolute value changes the sign, so the inequality becomes:
-(x - 2/3) ≤ 4
Next, we simplify this inequality:
-x + 2/3 ≤ 4
To isolate x, we'll subtract 2/3 from both sides:
-x ≤ 4 - 2/3
-x ≤ 10/3
Then, multiplying both sides of the inequality by -1 switches the direction of the inequality sign:
x ≥ -10/3
Thus, the solutions for the inequality |x - 2/3| ≤ 4 are:
Case 1: x ≤ 14/3
Case 2: x ≥ -10/3
To write the solution in interval notation, we combine the two cases:
(-∞, -10/3] ∪ [14/3, ∞)