Answers and Step-by-step explanations:
9. | x + 8 | ≤ 4
Remember that absolute value just denotes the distance between the argument within the absolute value signs and 0 on a number line. That means that if the item within the absolute value is negative, the absolute value of it will be the positive value because distance is always positive.
That means we have 2 cases here: x + 8 > 0 (positive) or x + 8 < 0 (negative).
Case 1: x + 8 > 0
x + 8 ≤ 4
x ≤ -4
Case 2: x + 8 < 0
-(x + 8) ≤ 4
-x - 8 ≤ 4
-x ≤ 12
x ≥ -12
Combining these two inequalities, we get:
-12 ≤ x ≤ -4
10. | x - 8 | + 5 ≥ 11
Let's isolate the absolute value expression by subtracting 5 from both sides: | x - 8 | ≥ 6
We again have two cases: x - 8 > 0 and x - 8 < 0.
Case 1: x - 8 > 0
x - 8 ≥ 6
x ≥ 14
Case 2: x - 8 < 0
-(x - 8) ≥ 6
-x + 8 ≥ 6
-x ≥ -2
x ≤ 2
Combining these two inequalities, we get:
x ≤ 2 and x ≥ 14
