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

Question

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

written in interval notation

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

written in interval notation

thanks!

Answer 1

1.

| 4 - 3x | + 1/2 &lt; -2

| 4 - 3x | + 1/2 - 1/2 &lt; -2 -1/2

| 4 - 3x | &lt; -2,5

S = { }

There is no such number whose absolute value is negative.

2.

| 3 - 2x | - 8 &gt;= 1

| 3 - 2x | - 8 + 8 &gt;= 1 + 8

| 3 - 2x | &gt;= 9

EITHER;

3 - 2x &gt;= 9

3 - 2x - 3 &gt;= 9 - 3

-2x &gt;= 6

-1/2*(-2x) &lt;= -1/2*6

x &lt;= -3

OR;

3 - 2x &lt;= -9

3 - 2x - 3 &lt;= -9 - 3

-2x &lt;= -12

-1/2*(-2x) &gt;= -1/2*(-12)

x &gt;= 6

S= (-∞,-3] U [6,∞)

Answer 2

Hi,

1. | 4 - 3x | + 1/2 &lt; -2

| 4 - 3x | &lt; -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!! :-)

Answer 3

|a|&lt;b means -b&lt;a&lt;b

|a|&gt;b means a&lt;-b or a&gt;b

so:

1. | 4 - 3x | + 1/2 &lt; -2

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

2. |3 - 2x | - 8 &gt;= 1

    |3-2x|&gt;=9

-9&gt;=3-2x or 3-2x&gt;=9

-12&gt;=-2x or -6&gt;=2x

x&gt;=6 or x&lt;=-3