Solve the absolute value inequality:?

Question

Problem 1.

|4y+3| -8 >= -3


Problem 2.

|x| >5

Answer 1

[1]
|4y + 3| - 8 ≥ -3
|4y + 3| ≥ 5
±(4y + 3) ≥ 5
4y + 3 ≥ 5 or -4y - 3 ≥ 5
4y ≥ 2 or -4y ≥ 8
y ≥ 1/2 or y ≤ -2
y is any real number in either interval,
(-∞, -2) or (1/2, ∞).

[2]
|x| > 5
±x > 5
x > 5 or -x > 5
x > 5 or x < -5.
x is any real number in either interval,
(-∞, -5) or (5, ∞).

Three things to remember:

First, flip-flop any inequality whenever you multiply or divide by a negative number.

Second, with compound inequalities: "OR" statements mean EITHER condition can be true... "AND" statements mean BOTH conditions MUST be true. Or's can have holes in the middle of their graphs, And's can't. Or's can become the set of all real numbers easily, And's can't. And's can have endpoints on both ends of a graph, Or's almost never do.

Third, when dealing with absolute value inequalities, when the absolute value is on the left side, greater-thans are OR statements and less-thans are AND statements. Think "greatOR" or "less-thAND."

[Edited to comment] Maybe_im_right08 and I are the only two with the correct solutions, so far. Trust us!

Answer 2

Problem 1.

| 4y + 3 | - 8 >= -3

| 4y + 3 | >= 5

4y + 3 >= 5
4y >= 2
y >= 1/2

- 4y - 3 >= 5
- 4y >= 8
y <= -2

Answer: y >= 1/2 or y <= -2

Problem 2.

| x | > 5

x > 5

- x > 5
x < -5

Answer: x > 5 or x < -5

Answer 3

|4y+3|-8>=-3
4y+3>=5 or 4y+3<= -5
4y>=2 4y<= -8
y>=1/2 y<= -2
so, -2>=y>=1/2

|x|>5
x>5 or x<-5
so, -5>x>5

Answer 4

1) |4y + 3| >= 5

so 4y+3 = 5 or -5 so y = 1/2 or -2

2) x = 5 or -5

Answer 5

1. y = 1/2 or (-2)
2. x = 5 or (-5)

Answer 6

Problem 1: 4y must be between -2.625 and -3.375, so the answer is "-0.84375 ≤ y ≤ -0.65625".

Problem 2: x < -5, or x > 5.

EDIT: Whoops, for problem 1, I thought "-8" meant "times negative 8", instead of "minus 8". In that case, the answer is -2 ≤ y ≤ 0.5

Answer 7

|4y + 3| - 8 >= -3
|4y + 3| >= 5
4y + 3 >= 5 or 4y + 3 <= -5
4y >= 2 or 4y <= -8
y >= (1/2) or y <= -2

y <= -2
or
y >= (1/2)

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|x| > 5
x > 5 or x < -5

Answer 8

Y = -1/2
X = 5 or -5

Answer 9

4y+3=_+5
y=-2
y=1/2

x>5
x<-5

Answer 10

p1) y>= 1/2 or y<= -2

p2) x>5 or x< -5