Help with absolute-value equations and inequalities?

Question

I've been trying to work on these problems all day, and I just can't figure them out! Here they are. Thank you.

7 | q | - 2 = 9

5 - 2 | 3x - 4 | = -5

| x | < 2 I think this one is { x | -2 < x < 2 } ?

| x - 1| < 4

| x + 2 | is smaller than or equal to 6

| x - 3 | + 2 > 7

| 2y - 9 | > -5

Answer 1

7|q| - 2 = 9
7|q| = 9 +2
7|q| = 11
q = 11/7 or -(11/7)

5 - 2|3x-4| = -5
-2|3x-4| = -10
|3x - 4| = -10 / -2
|3x-4| = 5
-3x +4 = 5 and 3x -4 = 5
-3x = 1 and 3x = 9
|x| = -(1/3), 3


you are right with number 3
|x| < 2
x < 2 and x > -2
-2 < x < 2


|x-1| < 4
x - 1 <4 and -x + 1 > 4
x <5 and x > -3
-3 < x < 5

|x + 2| <= 6
x + 2 <= 6 and -x - 2 <=6
x <= 4 and x >= -8
-8 <= x <= 4

|x - 3| + 2 > 7
|x - 3| > 5
x - 3 > 5 and -x + 3 < 5
x > 8 and x < -2


|2y - 9| > -5
2y - 9 > -5 and -2y + 9 < -5
2y > 4 and -2y < -14
y > 2 and y < 7
2 < y < 7

Answer 2

7Iq)-2=9 so 7IqI=7 IqI=1 and q=+-1
2I3x-4I=10 I3x-4I=5 3x-4 =+-5 and x= ( 4+-5)/3 (two values)
3) you are right
4) -4< x-1 <4 sum 1 to each term -3 5) -6<=x+2<=6 sum -2 -8<=x<=4
Ix-3I+2>7 ===> Ix-3I>5 You must split into x-3>5 so x>8

and x-3<-5 so x<-2

6) This is a tricky one because absolute value is always >=0
so it is always greater than a negative numberThe solution is all y

I hope that now you can do by yourself

Answer 3

for the first one, q= {11/7, -11/7}
for the second one,
3x-4= 5 or 3x-4=-5
so, x= {3, -1/3}
your answer for the third one is correct.
for the fourth one, -3 for the fifth one, -8 for the sixth one, actually I'm tired of this. Basically you consider the two cases where everything in the absolute value sign is either the number it says or the opposite of the number. In case of inequality, just use your common sense and see which sign to use.

Answer 4

7 |q| - 2 = 9
7 |q| - 2 + 2 = 9 + 2
7 |q| = 11
7 |q|/7 = 11/7
|q| = 11/7

6 - 2|3x-4| = -5
6 - 2|3x-4| - 6= -5 -6
-2|3x - 4| = -11
-2|3x-4|/-2 = -11/-2
|3x - 4| = 11/2

NOW, split it up into two equations.

3x - 4 = 11/2 and 3x-4 = - 11/2
3x - 4 +4= 11/2 + 4 and 3x-4+4 = -11/2 + 4
3x = 19/2 and 3x = -3/2
3x/3 = (19/2)/3 and 3x/3 = (-3/2)/3
x = 3.17 and x = -.5

You can probably figure out the rest.