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Solving equations and inequalities with absolute values

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by bdevas01 » Mon Feb 01, 2010 7:56 am
I'm a little confused when it comes to equations and inequalities containing absolute values.

As I understand it, the first step with both equations and inequalities is to isolate the absolute value. But absolute value problems usually have two solutions: one positive and one negative.

So what should be the next step? Should the value opposite the absolute value be negated? Or should the absolute value itself be negated? Is there different method for inequalities and equations?
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by ajith » Mon Feb 01, 2010 8:12 am
bdevas01 wrote:I'm a little confused when it comes to equations and inequalities containing absolute values.

As I understand it, the first step with both equations and inequalities is to isolate the absolute value. But absolute value problems usually have two solutions: one positive and one negative.

So what should be the next step? Should the value opposite the absolute value be negated? Or should the absolute value itself be negated? Is there different method for inequalities and equations?
One obvious answer to the questions posed is, "it depends". There is no one size fits all approach apart from a few ground rules.

Post the question you are facing the problem with a specific query and I am sure many in this forum will be more than happy to help you.
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by bdevas01 » Mon Feb 01, 2010 8:36 am
That's a good point Ajith. However, I still believe there are a few hard and fast rules when approaching absolute values.

Anyway, here are a few examples:

In this inequality, the absolute value itself is negated to find the negative solution:

|4x - 4| > 16

Positive:

4x - 4 > 16
4x - 4 + 4 > 16 + 4
4x > 20
x > 5

Negative:

-1(4x - 4) > 16
-4x + 4 > 16
-4x + 4 - 4 > 16 - 4
-4x > 12
x < -3

In this equation, the value opposite the absolute value expression is negated to find the negative solution:

|x + 5| + 20 = 60

|x + 5| + 20 - 20 = 60 - 20

|x + 5| = 40

X is Positive:
x + 5 = 40
x + 5 - 5 = 40 - 5
x = 35

X is Negative:
x + 5 = -40
x = -45

My question is, why is the method different for both problems?
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by ajith » Mon Feb 01, 2010 8:58 am
bdevas01 wrote:That's a good point Ajith. However, I still believe there are a few hard and fast rules when approaching absolute values.

Anyway, here are a few examples:

In this inequality, the absolute value itself is negated to find the negative solution:

|4x - 4| > 16

Positive:

4x - 4 > 16
4x - 4 + 4 > 16 + 4
4x > 20
x > 5

Negative:

-1(4x - 4) > 16
-4x + 4 > 16
-4x + 4 - 4 > 16 - 4
-4x > 12
x < -3

In this equation, the value opposite the absolute value expression is negated to find the negative solution:

|x + 5| + 20 = 60

|x + 5| + 20 - 20 = 60 - 20

|x + 5| = 40

X is Positive:
x + 5 = 40
x + 5 - 5 = 40 - 5
x = 35

X is Negative:
x + 5 = -40
x = -45

My question is, why is the method different for both problems?
To answer your query, if you are dealing with an equation it doesnt matter which side you negate (as in your second example) as long as you have only absolute function in one side. This is because

x = 1 means the same as -x = -1




But if you are dealing with an inequality, you need to be a bit more careful, because sign changes when you multiply both sides by a negative value.

What you need to do is to consider both possibilities without changing the sign (as you have done in the second example)

this is because

x<1 does not imply -x<-1
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by bdevas01 » Mon Feb 01, 2010 10:36 am
Yes, I think you're correct about how it doesn't matter which side is negated with equations.

As for inequalities, I believe the rule is that the absolute value expression is always the side that is negated. I've tried a few examples (posted below) and it appears to be the only way to get the right answer.

Example 1:

|x + 7| < 14

Positive:

x + 7 < 14
x + 7 - 7 < 14 - 7
x < 7

Negative:

-1(x + 7) < 14
-x - 7 < 14
-x - 7 + 7 < 14 + 7
-x < 21
x > -21

Example 2:

10 + |2x + 15| < 55

Positive:

2x + 15 < 45
2x + 15 -15 < 45 - 15
2x < 30
x < 15

Negative:

-(2x + 15) < 45
-2x - 15 < 45
-2x - 15 + 15 < 45 + 15
-2x < 60
x > -30

Example 3:

|x + 7| < 14
Positive:

x + 7 < 14
x + 7 - 7 < 14 - 7
x < 7

Negative:

-1(x + 7) < 14
-x - 7 < 14
-x - 7 + 7 < 14 + 7
-x < 21
x > -21
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by ajith » Mon Feb 01, 2010 10:39 am
Thanks for the examples. I think you summed up the discussion very well.
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