Question

Solve both equations the lines are absolute value and please include detailed steps for each

Answer 1

Answer:

see below

Step-by-step explanation:

|x+8| ≤ 4

Separate into two equations one positive and one negative , remembering to flip the inequality for the negative

x+8 ≤ 4        x+8 ≥ -4

Subtract 8 from each side

x+8 -8 ≤ 4-8    and     x+8-8 ≥ -4-8

x ≤-4     and    x ≥ -12

Writing as one function

-12 ≤x ≤-4

|x-8|+5  ≥ 11

Subtract 5 from each side

|x-8|+5-5  ≥ 11-5

|x-8|  ≥ 6

Separate into two equations one positive and one negative , remembering to flip the inequality for the negative

x-8  ≥ 6      or      x-8 ≤  -6

Add 8 to each side

x-8+8  ≥ 6+8       or     x-8+8 ≤  -6+8

x  ≥ 14      or      x ≤  -2

Answer 2

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