dddan 3 years ago help how do u solve the compound inequality; 1/2(x+6)>3 or 4(x-1)<3x-4

1. scarydoor

The two inequalities represent a set of values that x can take to satisfy the equation. For example all points satisfying x>1 is obviously (1,infinity). Treat each of the two separately. Because they are separated by "or" it means that you can take the union of the two sets, since picking any x from either of the two sets will satisfy at least one. To solve each separately, you need to rearrange to get just x on one side and everything else on the other. You can do this by knowing that you can add the same number to both sides, and you can multiply by a positive (non-zero) number to both sides, and you can multiply by a negative number, but then you'll need to swap the inequality. For example 3x>-4 is exactly equivalent to -3x<4.

2. dddan

i did that and got x>0 or x<0.....

3. scarydoor

For example, the left inequality: 1/2(x+6)>3. Multiply each side by 2: 2/2(x+6)>2*3. (x+6)>2*3. add -6 to both sides: x+6-6>2*3-6. x>2*3-6. x>0. So all points satisfying x>0 will make the left side true. And those are the only points, since you only used manipulations that left the equations exactly equivalent at each step.

4. scarydoor

Right side: 4(x-1)<3x-4. add -3x to each side: x-4<-4. add 4 to each side: x<0.

5. scarydoor

So you were right. But now you need all points that satisfy at least one of the equations. So if x>0 it satisfies the left. x<0 satisfies the right. So you can pick almost any x. .... except 0.

6. dddan

so how would i write the notation?

7. scarydoor

There are a few ways. You can write R \ {0}. Where you write the R like here: http://en.wikipedia.org/wiki/Real_numbers . This means "the set of all numbers, excluding the set containing the element 0". ("\" means "exclude all these things to the right. Kind of like subtraction). You could also write as intervals: (-infinity, 0) U (0,infinity). They mean the same thing but are just different ways of writing it.

8. dddan

thanks