Expert: Josh - 9/23/2004

Question
I have a question about how to solve this problem:

The absolute value of 5/9y-2/3 = -4

|5/9y-2/3| = -4

Answer
Hi Audrey,

For expressions like this, there are two cases to consider.

Here are the rules. If the quantity "a" (pick any number you like), inside the vertical bars |.| is positive, then, |a| = a. If the quantity "a" itself is negative, then, |a| = -a.

Key thing to remember: No matter how complicated the expression is, between the vertical bars, if the expression is positive, we don't change it. If the expression evaluates to some negative number, we introduce a minus sign to make it positive.

But I doubt that the question has been set correctly. Because the absolute value of any real quantity is always positive.

In fact an expression like |x-m|=d is a geometric measurement of distance. It refers to two points on an interval R units from the point x=m. For example, if the midpoint m=3, and the distance d=1, you will find that |x-3|=1 has solutions x=2,4. And their distance is exactly d=1 unit from x=3. It makes no sense turning the right hand side, "d", into a negative number.

Returning to the question now, let's say that we have to solve |5/9y-2/3| = 4 for y.

Case 1:
If the left hand side (LHS) expression is non-negative, then,
|(5/9)*y-(2/3)| = 4 is equivalent to
(5/9)*y-(2/3) = 4 ...can safely remove the vertical bars,
(5/9)*y = (2/3)+4
y = (9/5)*[(2/3)+4] ...[*]

Case 2:
If the left hand side (LHS) expression is negative, then,
|(5/9)*y-(2/3)| = 4 is equivalent to
-[(5/9)*y-(2/3)] = 4 ...we have introduced the minus sign
(2/3)-(5/9)*y = 4
(5/9)*y = (2/3)-4
y = (9/5)*[(2/3)-4] ...[**]

You can simplify the fractions if you like.

Understanding the expression |ay-b|=c:
Observe that |ay-b|=c is same as |a(y-b/a)|=c,
so, |a|*|y-b/a|=c, |y-b/a|=c/|a| ......[#1]

One way to interpret the equation [#1] is that the solutions for y represent points at a distance of c/|a| from y=(b/a). As an exercise, you can check this yourself using a=5/9, b=2/3, c=4, y=(9/5)*((2/3)+4) or (9/5)*((2/3)-4).

This is equivalent to saying that after finding the distance between y and b/a, scaling this distance by "a" gives you a distance of "c".

Let us verify this claim.
Here, a=5/9, b=2/3, c=4.
c/|a| = c/a = (2/3)*(9/5) = 18/15 = 1.2

Case 1: When y=(9/5)*((2/3)+4)=8.4,
Distance |y-c/a|=|8.4-1.2|=7.2
Scaling the y-axis by a factor of "a=5/9",
a*|y-c/a|=(5/9)*7.2=4 ...as required.

Case 2: When y=(9/5)*((2/3)-4)=-6,
Distance |y-c/a|=|-6-1.2|=|-7.2|=7.2
Scaling the y-axis by a factor of "a=5/9",
a*|y-c/a|=(5/9)*|-7.2|=4 ...as required.

Cheers.