# 2 Inequalities with Abs. Value - written in interval notation :)?

1. | 4 - 3x | + 1/2 < -2

written in interval notation

2. |3 - 2x | - 8 >= 1

written in interval notation

thanks!

Relevance

Hi,

1. | 4 - 3x | + 1/2 < -2

| 4 - 3x | < -2½

This is looking for the values of x such that the absolute value of 4 - 3x is less than -2½. Since an absolute value is nonnegative, it is NEVER less than a negative number. For this problem there is no solution, so the answer is Ø.

2. |3 - 2x | - 8 ≥ 1

|3 - 2x | ≥ 9

3 - 2x ≥ 9 or 3 - 2x ≤ -9

-2x ≥ 6 or -2x ≤ -12

x ≤ -3 or x ≥ 6

The answer is (-∞,-3) U (6,∞)

I hope that helps!! :-)

• |a|<b means -b<a<b

|a|>b means a<-b or a>b

so:

1. | 4 - 3x | + 1/2 < -2

|4-3x|<-5/2. I think it is impossible because |a| means the absolute value of a, so the things that are between the"|" is always positive.

2. |3 - 2x | - 8 >= 1

|3-2x|>=9

-9>=3-2x or 3-2x>=9

-12>=-2x or -6>=2x

x>=6 or x<=-3

• Anonymous

1.

| 4 - 3x | + 1/2 < -2

| 4 - 3x | + 1/2 - 1/2 < -2 -1/2

| 4 - 3x | < -2,5

S = { }

There is no such number whose absolute value is negative.

2.

| 3 - 2x | - 8 >= 1

| 3 - 2x | - 8 + 8 >= 1 + 8

| 3 - 2x | >= 9

EITHER;

3 - 2x >= 9

3 - 2x - 3 >= 9 - 3

-2x >= 6

-1/2*(-2x) <= -1/2*6

x <= -3

OR;

3 - 2x <= -9

3 - 2x - 3 <= -9 - 3

-2x <= -12

-1/2*(-2x) >= -1/2*(-12)

x >= 6

S= (-∞,-3] U [6,∞)