Absolute Values

[topaz horizon]

Hello there, I am having trouble understanding double absolute value equations and inequalities. Don’t have to do all questions from e to k just any one of them with proper working out not just the answer so I can compare working out. (struggling with questions e) to k)) only. It would be nice if I can get some explanation for these types of questions. Thanks.

[lusty reef]

,rotate

[lusty reef]

you can simply square both sides to get rid of the absolute

[topaz horizon]

I want to learn the other method tho

The one that uses 3 cases

[lusty reef]

|x| = x, if x > 0
|x| = -x, if x <= 0
this one?

[topaz horizon]

Yup

[lusty reef]

for example (a):
case 3x + 1 > 0:
3x + 1 = 2x + 4
3x - 2x = 4 - 1
x = 3
case 3x + 1 <= 0:
-(3x + 1) = 2x + 4
-3x - 1 = 2x + 4
-3x - 2x = 1 + 4
-5x = 5
x = -1
hence the solutions are
x = 3 and x = -1

[topaz horizon]

Wait no I said I am struggling from e) to k)

Like question i)

Sorry not e)

From k) onwards

Sorry

Can u explain question i with the three cases

Sorry

[opal mural]

Remove the absolute value bars and solve for x when 12 is positive and when 12 is negative

[lusty reef]

oh ok
case x - 3 > 0:
then x + 2 > 0
(x - 3) + (x + 2) = 4
2x - 1 = 4
2x = 5
x = 5/2
case x - 3 < 0:
if x + 2 > 0:
-(x - 3) + (x + 2) = 4
-x + 3 + x + 2 = 4
5 = 4 (no solution)
if x + 2 <= 0:
-(x - 3) - (x + 2) = 4
-x + 3 - x - 2 = 4
-2x + 1 = 4
-2x = 3
x = -3/2
hence the solutions are
x = 5/2 and x = -3/2

[opal mural]

My bad, what I said won’t work for all the problems. Swindle’s is more accurate

[topaz horizon]

Why did u use x+2<=0. How do I know when I am supposed to use <= or >= for cases

I don’t understand the last case and also for the first and second case why didn’t u use <= >= sign?

[lusty reef]

cuz this @topaz horizon

I just divide it case by case

[topaz horizon]

That means there’s no solution for this question

If u sub the x values in

[topaz horizon]

Wait if x = 0 how is |x| =-x When |x| =-x, if x<=0(meaning x can be equal to zero or less right in order for it to equal to -x)

If x itself is equal to zero shouldn’t |x| also equal 0?

[lusty reef]

cuz -0 = 0

[topaz horizon]

WAIT

I THINK I GOT IT

SOMIT DOESNT MATTER IF WE PUT >= <= either way cause we are trying to get rid of absolute value

Can I send u working out for one of the questions and can u check it

It’s question g)

Therefore x=3 and -3

That means question i) has no solution x can’t be equal to 5/2 or -3/2 cause they don’t satisfy the inequality of the cases

I was confused because of the resultant inequalities people use for cases. Now I get it they just use the cases but use final inequality that satisfies both for all three cases for these types of questions as it is quicker. For e.g for case 1 resultant inequality has to be x>= 3 as it satisfies x>=2 too.

So first case is x>=3 meaning it will be x-3+x+2=4 x=5/2 but 5/2 isn’t more than or equal to 3!!