Quadratic equations and inequalities

[Kartik.]

Well suppose for an example of an inequality,
|x-1|-|x|+|2x+3| > 2x+4

Well in one of its solutions we were told to apply the method of intervals, rather than taking say what; like 8 combination of signs.

For everyone of its intervals(say -3/2 $\leq$x <0) we are said that 2x+3 $\geq$ 0, x<0 and x-1<0.

Is it just an intuitive outcome(guessing by the extremities of the limits) or did we do something to predict what the signs of the expressions will be?(as i was wondering if any such intervals, which violated an intuitive outcome ?)

 

[HallsofIvy]

I'm not sure what you mean by "intuitive" but the method of "predicting" the sign is to use the definition of "absolute value": |x|= x if $x\ge 0$, -x if x< 0.

Of course that says that the "change" comes when the argument of the absolute value function is 0. In order to deal with |x-1|-|x|+|2x+3| > 2x+4, I would note that x-1= 0 when x=1, x= 0 when x= 0, of course, and 2x+3= 0 when x= -3/2. Now put those in order: -3/2< 0 < 1. So if x< -3/2, it is also less than 0 and 1 and all of x-1, x and 2x+3 are negative. For x< -3/2, the inequality becomes -(x- 1)-(-x)+ (-(2x+3))> 2x+ 4. That is the same as -x+ 1+ x- 2x- 3= -2x- 2> 2x+ 4 which is, in turn, equivalent to -6> 4x or -3/2< x. Of course, if x< -3/2, "-3/2< x" can't be true so there is NO value of x, less than -3/2, satisfying that inequality.

If $-3/2\le x< 0$, x and x- 1 are still negative but 2x+3 is now non-negative and so the inequality is -(x-1)-(-x)+ (2x+3)> 2x+ 4. That is the same as -x+ 1+ x+ 2x+ 3= 2x+ 4> 2x+4 which is, again, never true (they equal not "larger than"). There is no x between -3/2 and 0 satifying this inequality.

If $0\le x< 1$ both 2x+3 and x are positive but x- 1 is still negative. Now the unquality is -(x-1)- x+ (2x+3)> 2x+ 3. That is the same as -x+ 1+ x+ 2x+ 3= 2x+ 4> 2x+ 3 which is always true. Every value of x, $0\le x< 1$ satisfies this inequality.

Finally, if $x\ge 1$ all three of x-1, x, and 2x+3 are positive. Now the inequality is (x- 1)- x+ (2x+ 3)= 2x+ 2> 2x+ 3. That is never true.

Putting all of those together, the original inequality is satified for all x, $0\le x< 1$ and no other x. The solution set is the interval [0, 1).