Equations with Absolute Value

by Amy McKain
(Brazil, IN)

Here is the question. It is 3 absolute value of x divided by 9 plus 7 equals 8.

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Equations with Absolute Value

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Jan 25, 2011
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by: Karin

You solve this equation very similar to a regular equation with the absolute value.

We are going to address the absolute value in the last step.

The problem:
((3|x|)/9) + 7 = 8

Step 1: Subtract 7 from both sides.

((3|x|)/9) +7-7 = 8 -7

((3|x|)/9) = 1

Step 2: Get rid of the fraction by multiplying all terms by 9.

9((3|x|)/9)=1(9)

3|x| = 9

Step 3: Divide by 3 on both sides.

(3|x|)/3 = 9/3

|x| = 3

Now we are left with the absolute value of x = 3. Remember that absolute value means that the value of the number inside of that absolute value sign is always positive.

So we can say that:
x = 3
or -x = 3

If -x =3 then we would multiply all terms by -1 to make x positive.

-1(-x) = 3(-1)
x = -3

Basically you know that 3 or -3 could be inside of the absolute value symbol and the value of that number will always be 3.

Check your answer:

3|3|/9 + 7 = 8

9/9 + 7 = 8
1+7 = 8

So, we know that 3 works.

Now check -3

3|-3|/9 + 7 = 8
3(3)/9 + 7 = 8

9/9 + 7 = 8
1+7 = 8

-3 also works. Therefore, x could be equal to 3 or -3.

x = 3
x = -3

I hope this helps,
Karin

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