inequalities

Solve x-3=0 and 2x+1=0, which are x=3 or x=-1/2
If x<-1/2. Then |x-3| +|2x+1| <4 becomes
3-x-(2x+1)<4
2-3x<4
-2<3x
x>-2/3
So -2/3<x<-1/2

If -1/2<=x<3. Then |x-3| +|2x+1| <4 becomes
3-x+(2x+1)<4
x+4<4
x<0
So -1/2<=x<0

If x>=3. Then |x-3| +|2x+1| <4 becomes
x-3+(2x+1)<4
3x<6
x<2
which is a contradiction

So, the answer is -2/3<x<0

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

case 1: x<-1/2
then x-3<0, 2x+1<0
so
-(x-3)-(2x+1)<4
-x+3-2x-1<4
-3x<2
x>-2/3
∴-2/3<x<-1/2

case 2: -1/2≦x<3
then x-3<0, 2x+1≧0
so
-(x-3)+(2x+1)<4
-x+3+2x+1<4
x<0
∴-1/2≦x<0

case 3: 3≦x
then x-3>0, 2x+1>0
so
x+3+2x+1<4
3x<0
x<0
which is never true
∴no solution

combining the three case, the overall solution is
-2/3<x<0

Example 1:Consider the inequality


The basic strategy for inequalities and equations is the same:
isolate x on one side,
and put the "other stuff" on the other side. Following this strategy,
let's move +5 to the right side.
We accomplish this by subtracting 5 on both sides (Rule 1) to obtain

after simplification we obtain

Once we divide by +2 on both sides (Rule 3a),
we have succeeded in isolating x on the left:

or simplified,

All real numbers less than 1 solve the inequality. We say that the "set of solutions'' of the inequality consists of all real numbers less than 1. In interval notation, the set of solutions is the interval .

Example 2:Find all solutions of the inequality

Let's start by moving the ``5'' to the right side by subtracting 5 on both sides (Rule 1):

or simplified,

How do we get rid of the ``-'' sign in front of x? Just multiply by (-1) on both sides (Rule 3b), changing " " to " " along the way:

or simplified

All real numbers greater than or equal to -1 satisfy the inequality. The set of solutions of the inequality is the interval .

Example 3:Solve the inequality

Let us simplify first:

There is more than one route to proceed;
let's take this one: subtract 2x on both sides (Rule 1).

and simplify:

Next, subtract 9 on both sides (Rule 1):

simplify to obtain

Then, divide by 4 (Rule 3a):

and simplify again:

It looks nicer, if we switch sides (Rule 2).

In interval notation, the set of solutions looks like this: .

Exercise 1:Find all solutions of the inequality


Answer.

Exercise 2:Solve the inequality


Answer.

Exercise 3:Solve the inequality


Answer.

Exercise 4:Find all solutions of the inequality


Answer.

2009-11-03 12:14:03 補充：
Rule 1. Adding/subtracting the same number on both sides.