Solving Absolute ValueEquations & InequalitiesA step-by-step guide to Absolute Value Problems

To put it simply.

What does absolute value mean and why is it important?

There is a technical definition for absolute value, but you could easily never need it. For now, you should view the absolute value of a number as its distance from zero.

Think of it as a number line…Let's look at the number line:

The absolute value of x, is the distance of x from zero.

• This is why absolute value is never negative; • Absolute value only asks "how far?", not "in which

direction?". • This means not only that | 3 | = 3, because 3 is three

units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.

1 2 3 4 5 0

0

1

1

2

2

3

3

4

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5

5

– 1

– 1

– 2

– 2

– 3

– 3

– 4

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– 5

– 5

The symbols

Absolute value has a symbol, actually two, just like other operations.

The symbols for absolute value are two vertical lines. They are meant to surround the value that you want to take the absolute value of, sort of like parenthesis surround the symbols that they group.

An example.

Here are two simple examples. Say that I wanted to take the absolute value of -5. I would write it like this:

-5

This would be read in English as, “The absolute value of negative 5.”

We got it? Here’s a few more.

|-6| The absolute value of negative 6.

|10|

|y|

|x|

|-y +2|

|0|

The absolute value of 10.

The absolute value of x.

The absolute value of y.

The absolute value of negative y plus 2.

The absolute value of 0.

Absolute value in action.

So what are the answers? What is the absolute value of negative 5 equal to?

-5 5=

Five!

How it works for all numbers (inputs)

It’s simple. Well, it’s a simple as this:

If an input is positive, it STAYS positive.If an input is negative, it becomes positive.If an input is zero, it stay zero.

Got it? Try to apply it.

|-6| = 6.

|10|

|y|

|x|

|-y +2|

|0|

= 10. Note: NOT negative 10. Taking the absolutevalue is NOT the same as taking the opposite.

= x. But note, we still don’t know what x is.

= y. y might be negative, positive, or zero.

This would have to be graphed. Y can be anythingand then we would shift the graph 2 to the right.

= 0. The absolute value of 0 is 0. Period, endof story.

Stay with me, there’s more.

Ok, we now know what absolute value does, but if that’s a new concept to you then practice it well. To reach the level of the standard we have to move on.

First lets look at a simple equation and solve it:

x + 10 = 293 -10 = -10

x = 283Subtract 10 from both sides.Solution x = 283.

I hope that doesn’t shock anyone. If it does please go back and review basic algebra. The rest of this will only confuse you if you don’t.

Now a little thinking.

Let’s add absolute value into this same equation:

|x + 10| = 293

This should be read: “The absolute value of x + 10 equals 293.

Now we just saw that 283 is the answer to this problem and I will tell you that it is the ONLY solution. That is it is the only replacement for x that makes the statement x + 10 = 293 a true statement.

Another story

With absolute value in the equation:

|x + 10| = 293

Let’s think. What if x + 10 came out to be -293.

Then we would have |-293| = 293.

And that’s a true statement.

Think even harder.

-303 + 10 equals = -293So if x equaled -303 then the equation

would be true.There are TWO solutions to the equation |

x + 10| = 293.In fact there usually are two solutions to

an equation that involves absolute value.

The good and the bad.

And I have good news and bad news.

The good news is that you don’t have to GUESS every time you encounter an absolute value problem.

The more good news is that there is a systematic method for finding both solutions.

The bad news is that you will have to learn and memorize this method.

The method

First isolate the absolute value sign on one side:It has to read, “The absolute value of something,

equals something.”With our sample problem we’re already good.Now you have to change the right side of the

equation and get rid of the absolute value signs. We are going to have two solutions and so we’re going to have two equations.

|x + 10| = 293

We have:

x + 10 = 293 x + 10 = - 293and:

Seem strange?

That’s right we have:

x + 10 = 293 x + 10 = - 293and:

It may seem strange to change the right side of the equation to find out what that the variable is on the LEFT side, but trust me it works.

Notice that the absolute value signs are now GONE. These two are easy to solve.

Two worked out solutions

x + 10 = 293 - 10 = - 10

x = 283

x + 10 = -293 - 10 = - 10

x = - 303

Seem strange?

We get two solutions.

x = 283 x = - 303and:

This may seem strange but they both make the original equation true. Watch…

| x + 10 | = 293Plug in 293…..|283 + 10| = 293| 293 | = 293 293 = 293 true

Now the other one.

| x + 10 | = 293Plug in -303…|-303 + 10| = 293| -293 | = 293 293 = 293 true

See? This one works too.

Let’s review.

Remember. When the absolute value signs get involved in an equation then you can expect that there will be TWO solutions and constructing TWO equations is necessary to finding these solutions.

Isolate the absolute value on one side of the equation.

Make two versions of the equation. In one make the NON-absolute value side negative, in the other make it positive.

What about > < < >

The standard demands that we also deal with inequalities.

Inequalities are also mathematical statements. That is, they SAY something about the relationship between these numbers. And just like when a person says something, what they say may be true or it may be false.

Inequalities do NOT make the simple statement that one side is equal to the other. Inequalities can say one of four things:

_ _

What can they say?

> Says that the left side is GREATER THAN the right side.

< Says that the left side is LESSER THAN the right side.

≥ Says that the left side is GREATER THAN OR EQUAL to the right side.

≤ Says that the left side is LESSER THAN OR EQUAL to the right side.

Examples: 1 < 3 Reads: “One is less than three”, a true

statement.4 > -2 Reads “Four is greater than negative

two”, and is also a true statement.

More examples:

More examples:◦ 4 ≤ 4 This says, “four is lesser than or equal to four.” a

true statement.◦ 8 ≥ 8 This says, “eight is greater than or equal to eight.”

a true statement.Let’s say some FALSE THINGS just for fun:

◦ 8 ≥ 19. This is read: “Eight is greater than or equal to 19.” but 8 is not greater than or equal to 19, so this is false.

◦ 9 < -10 This is read: “Nine is less than negative ten”. Negative numbers are inherently less than positive numbers. This is false.

The new rule:

Inequalities are EASY to solve if you know how to solve regular equations. There is just one new rule that you have to remember:

If you multiply or divide by a negative number your must turn the inequality sign towards the other direction. This flips its meaning.

True for all inequalities:

This is true for inequalities whether there is an absolute value sign in the inequality or not.

Absolute value and Inequality

Now before we get into truly tackling an absolute value inequality we have to talk about a rather complex behavior that happen when you combine absolute value with an inequality sign.

I’m not going to lie to you, this procedure is pretty tough so listen very carefully, or play this part of the video over and over until you get this.

Absolute value can best be understood as: “The distance that something is from zero on a number line.”

Let’s start with a very simple absolute value inequality:

0

x > 3

x > 3

Let’s read this in a way that will help us draw it on a number line.It says: “Whatever x is, it must be more then 3 spaces away from 0 on anumber line.”

0

So how do we make that happen?

Read it right to get it right.

x < 3

What if this had a lesser than sign, instead of a greater than sign?Then it would say: “Whatever x is, it must be less than 3 spaces away from 0 on a number line.”

0

How do we make that happen?

Read it right to get it right.

Please help.

Now, since this is tough to memorize. Try this little poem or make up one of your own.

“If the sign is greater than see you later.”“If the sign is less then you shouldn’t

stress just stay inside and clean up your mess.”

Action time!

OK, pause the video here and let that set in.

|-3y -8| + 10 > 100

Can you read this now?

It says: “The absolute value of the quantity negative 3 times y minus 8, plus 10 is greater than 100.”

If you’re still here, we are moving on.

Consider this example inequality:

Isolate the absolute value

Remember that our first task is to get the absolute value to be on it’s own on one side of the inequality.

|-3y -8| + 10 > 100

So in this case what needs to be dealt with?

That’s right!! the + 10.How do we get rid of a plus 10?That’s right!! We subtract 10.

Chugging through the first algebra.

|-3y -8| + 10 > 100 - 10 -10 | -3y – 8 | > 90

So far so good. Now we have to break this into two related inequalities. What was my terrible poem again?

“If the sign is greater than see you later.”“If the sign is less then you shouldn’t stress just stay inside and clean up your mess.”

|-3y -8| > 90

That means we will have one inequality that just gets rid of the absolute value signs.

And one that gets rid of the absolute value sign and:◦ Flips the inequality symbol to the other direction.◦ And changes the sign of the right hand side.

-3y -8 > 90 -3y -8 < -90

becomes:

and…

Create two related inequalities

Here we have “GREATER THAN” so we are going to “See you later man”. There we are going to send our arrows to the right and left.

-3y -8 > 90

Recall that we use inverse operations to solve, with the goal being to get y by itself.

Add 8 to both sides to get rid of the -8. + 8 > + 8

-3y > 98 Now divide by negative 3 and flip the in-equality sign as you do so. This is necessary to find the correct solution. Do not forget it.

/ -3 > / -3

y < -32.6

Y is less then negative -32.6 is our answer.

Let’s work the first inequality that we created.

-3y -8 < -90

We are not done. Now we work our second equation to get our second answer.

+ 8 < + 8

-3y < -82 Now divide by negative 3 and flip the in-equality sign as you do so. This is necessary to find the correct solution. Do not forget it.

/ -3 < / -3

y > 27 1/3

Y is greater than 27 1/3 is our answer.

Add 8 to both sides to get rid of the -8.

Graph it.

Graph our solutions

Let’s read this in a way that will help us draw it on a number line.It says: “Whatever x is, it must be more then 32 2/3 spaces away from 0 on a left side of 0 and more that 27 1/3 spaces away from 0 on the right of the number line.”

0-20 -10 10 3020-30

y < -32 2/3 y > 27 1/3 OR

Now let’s check our solutions but remember that we have to pick a number slightly different then our solution or boundary numbers.

One of the things that we have said is that our y for this problem is:

Pick a value < -32 2/3

y < -32 2/3

Y is less then negative 32 2/3.What number can we pick the will be just a tiny

bit less then negative 32 2/3?How about……-33I like it.

Now let’s check our solutions but remember that we have to pick a number slightly different then our solution or boundary numbers.

The other thing that we have said is that our y for this problem is:

Pick a value > 27 1/3

y > 27 1/3

Y is greater than negative 32 2/3.What number can we pick the will be just a tiny

greater than negative 27 1/3?How about……27.5I like it.

Check for TRUTH not Equality.

Let’s checkPlug BOTH of these choices in for y and then

work the math to see for sure that you get down to a TRUE statement.

Remember that they do not need to be EQUAL.Our solutions do NOT say that they should be

equal. They should make are statements tell the truth. If

our statement said it would come out less than then it should come our less then, and if our statement said that it should come out greater than then it should come out greater than.

|-3y -8| + 10 > 100

|-3( ) – 8 | + 10 > 100

|-3(-33) – 8 | + 10 > 100

|99 – 8 | + 10 > 100

| 91 | + 10 > 100

91 + 10 > 100

101 > 100

Here is our original.

Put parenthesis in place of Y.

Plug in -33 for YNotice I did not removethe absolute value thistime. I am NOT followingthe solution steps thatI showed you earlier here.I am just running throughthe math and checking myanswer.

The absolute value of 91 is 91, so now the absolute val signs go away.

|-3y -8| + 10 > 100

|-3( ) – 8 | + 10 > 100

|-3(27.5) – 8 | + 10 > 100

|-82.5 – 8 | + 10 > 100

| -92.5 | + 10 > 100

92.5 + 10 > 100

102.5 > 100

Here is our original.

Put parenthesis in place of Y.

Plug in 27.5 for YNotice I did not removethe absolute value thistime. I am NOT followingthe solution steps thatI showed you earlier here.I am just running throughthe math and checking myanswer.

The absolute value of -92.5 is 92.5, so now the absolute val. signs go away.

Pause and practice, but there’s more.

Excellent.

Thank you for hanging with me.

DO NOT be discouraged if you didn’t catch all of

that in the first go around.

Just re-play.

Ask questions in the comments

Send us emails at math@whaleboneir.com

|-2y -6| + 5 < 100

•We still have to consider a problem where the left side is less than the right side.

•Like this:

|-2y -6| + 5 < 100

•Notice that we are dealing with a lesser than sign here. So we refer back to our limerick: “If the sign is greater then ‘See you later.’ If the sign is less then just don’t stress, just stay inside and clean up the mess.”

0-20 -10 10 3020-30 1 2

•We are going to have a closed in answer here. The range of values that will make this true are going to be between one number and another number but won’t include those numbers.

•This is what our diagram might look like:

•And in set notation we might have:

{ -29 < y < 28 }

but these number are just guesses at this point.

•Let’s work it.

•Step 1: Get the absolute value sign alone on one side of the inequality:

Pause and practice, but there’s more.

|-2y -6| + 5 < 100 - 5 < - 5 | -2y – 6 | < 95

By subtracting 5 from both sides.

| -2y – 6 | < 95 Now we create two related inequalities out of this, which will allow us to get rid of the absolute value signs.

-95 < -2y – 6 and -2y – 6 < 95

-95 < -2y – 6 and -2y – 6 < 95 + 6 +6 +6 +6-89 < -2y and -2y < 101 44.5 > y y < -51 { -51 < y < 44.5 }

Check for TRUTH not Equality.

Let’s checkPlug BOTH of these choices in for y and then

work the math to see for sure that you get down to a TRUE statement.

Remember that they do not need to be EQUAL.Our solutions do NOT say that they should be

equal. They should make are statements tell the truth. If

our statement said it would come out less than, it should come our less than, and if our statement said that it should come out greater than then it should come out greater than.

|-2y -6| + 5 < 100 Our original

|-2( ) -6| + 5 < 100 Carefully put in parenthesis for y.

|-2(-50) -6| + 5 < 100 We need Y> -51. So I’ll chose -50.

|100 - 6| + 5 < 100 -2*-50 = 100

|94| + 5 < 100 100 – 6 = 94

94 + 5 < 100 The ABS of 94 is 94.

99 < 100 A true statement. This answer works.

|-2y -6| + 5 < 100 Our original

|-2( ) -6| + 5 < 100 Carefully put in parenthesis for y.

|-2(44) -6| + 5 < 100 We need Y< 44.5 So I’ll chose 44.

|-88 - 6| + 5 < 100 -2*-50 = 100

|-94| + 5 < 100 -88 – 6 = 94

94 + 5 < 100 The ABS of -94 is 94.

99 < 100 A true statement. This answer works.