how do we solve this Ix+3/2I≤1/2? And what do you call them in English ?

Answer 1

miinii almost had it right, it's AND not OR in this case.

That is: Ix+3/2I≤1/2 is the same as

x+3/2 ≤ 1/2

x ≤ -1
and
- 1/2 ≤ x+3/2
-2 ≤ x

-2 ≤ x ≤ -1

Answer 2

This is an absolute value inequality.
Absolute value is distance from zero on the number line.

That means x+3/2 must be less than or equal to 1/2 of a unit from 0. You can be a 1/2 unit to the right or a 1/2 unit to the left. So x+ 3/2 must fall in between -(1/2) and positive 1/2. This gives the compound inequality to solve: -(1/2)<=(x+3/2)<=1/2.

When you solve this you do get -2<=x<=-1

Answer 3

Option 1:
x+3/2 <= 1/2 and x+3/2 >= 0
x <= -1 and x >= -3/2
-3/2 <= x <= -1

Option 2:
-(x+3/2) <= 1/2 and x+3/2 <= 0
x+3/2 >= -1/2 and x <= -3/2
x >= -2 and x <= -3/2
-2 <= x <= -3/2

Take the union of the two ranges:
-2 <= x <= -1

They are called inequalities in English.

Answer 4

| x + 3/2| ≤ 1/2, called an absolute value inequality, becomes

x + 3/2 ≤ 1/2, x ≤ -1
or
-(x + 3/2) ≤ 1/2
x + 3/2 ≥ - 1/2
x ≥ -2

Answer 5

Either x + 3/2 <=1/2 and x + 3/2 >= -1/2
=> x <= -1/2 and x >= -2
thus x should lie between -2 and -1/2, both inclusive

Answer 6

first set up as x+3/2 ≤ 1/2
then solve the inequality. which will give u x ≤ -1
then set up as x+3/2 ≥ -1/2
solve the inequalty. which gives u x ≥ -2
so the answer is -2 ≤ x ≤ -1

Answer 7

| x + 3/2 | <=1/2
so
x+3/2<=1/2
x<=1/2-3/2
x<=-1

or

-(x+3/2)<=1/2
x+3/2>=-1/2
x>=-1/2+3/2
x>=-2

Answer 8

In my class I'm the best in maths, but this I can't solve it.