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Absolute Value Equations and Inequalities
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Absolute Value Equations and Inequalities
The absolute value of a number is simply “how big” the number is, without a + or - sign in front
of it. The technical definition is given on page 140 of the textbook, but the meaning is the
common one … e.g. |4| = 4 and |-4| = 4.
Since there are always two numbers with the same absolute value, solving absolute value
equations requires a little care. We use the same techniques and procedures used to solve
“normal” equations, but we have to check out two possible solutions.
We’ll look at three cases here: (1) solving and equation for the absolute value of a variable,
(2) solving an equation for the absolute value of a function of the variable, and (3) solving
inequalities with absolute values. The difference here will be that we’ll have to add a step,
since the absolute value can arise from two different numbers.
1. Solving equations for absolute value of variable:
This is straight forward: we solve the equation as we would any other equation, and then we
consider the two possibilities.
Let’s look at an example.
|3x| + 5 = 14
We want to isolate the variable and then look at the two possible answers.
|3x| + 5 - 5 = 14 - 5
|3x| = 9
One solution:
3x = +9
x=3
The other solution:
3x = -9
x = -3
Then our solutions are 3 and -3, and we can write the solution set as {3, - 3}.
2. Solving equations for the absolute value of a function of the variable:
Actually, we just did this in the previous example: “3x” is not the variable itself, but a function
of the variable (three times the variable). Let’s look at an example:
|4z - 2| + 8 = 46
First we isolate the absolute value term.
|3z -2| + 8 - 8 = 46 - 8
|3z - 2| = 38
Now we look at the two possibilities:
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6/23/2003
Absolute Value Equations and Inequalities
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First the “positive” case:
3z - 2 = 38
3z - 2 + 2 = 38 + 2
3z = 40
z = 10
And the “negative” case:
3z - 2 = -38
3z - 2 + 2 = -38 + 2 = - 36
3z = -36
z = -9
So our solution set is {10, -9}.
3. Absolute values on both sides:
This looks a little daunting at first, but the principle is the same … we consider the “positive”
case (make the right side positive) and the “negative” case (make the right side negative.
Let's look at an example:
|9y + 1| = |6y + 4|
When we consider a case, we discard the absolute value signs.
Let’s start with the right side positive:
9y + 1 = +(6y + 4)
9y + 1 = 6y + 4
9y + 1 - 1 = 6y + 4 - 1
9y = 6y + 3
9y - 6y = 6y - 6y + 3
3y = 3
y=1
That’s one solution. Now let’s make the right side negative, being careful to
distribute that negative sign:
9y + 1 = -(6y + 4)
9y + 1 = -6y - 4
9y + 1 - 1 = -6y - 4 - 1
9y = -6y - 5
9y + 6y = -6y + 6y - 5
15y = -5
y = -5/15 = -1/3
y = -1/3
The solution is then {1, - 1/3}.
Solving inequalities:
There is a special way to handle inequalities with absolute values … we write the absolute
value as a compound inequality and solve that. Let's look at an example:
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6/23/2003
Absolute Value Equations and Inequalities
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|3x| + 5 < 17
First, we need to isolate the absolute value expression:
|3x| + 5 - 5 < 17 - 5
|3x| < 12
Now we use the positive case … 3x < 12 … and the negative case … 3x > -12, or -12 < 3x …
to write a compound inequality.
-12 < 3x < 12
Solving this, we get -4 < x < 4 and the solution set is (4, 4).
http://66.20.58.22/courses/1/SUCCESS/content/_28527_1/Absol_value_eq_ineq.htm
6/23/2003