Working with Absolute Value

In mathematics, we use the concept of absolute value in two rather different looking but equivalent contexts.

1. In a geometric sense, the absolute value of x, written |x|, means:

|x| is the distance from x to 0 on the number line.

Then |6| = 6 and |-6| = 6

What if we don’t know the value of x? In other words, how can we use this geometric idea to define |x| in general? Notice that 6 and -6 both have absolute values of 6. This is because they are the same distance from 0 on the number line, only 6 is to the right of 0 and -6 is to the left of 0. Since distance is always nonnegative, |x| must always be nonnegative.

2. This suggests our second definition of |x| as:

|x| = x if x ≥ 0 and –x if x < 0.

This second definition is used frequently in solving equations and inequalities, while the geometric definition is often more comfortable in solving problems. We’ll look at both in some examples soon.

Let’s look at one more interpretation of absolute value, similar to our geometric one. What is the distance from 7 to 5 on the number line? That’s easy! 7 is 2 units away from 5. Just subtract 7 – 5, no problem! What is the distance from -1 to 5 on the number line? That’s not bad, either – just subtract 5 – (-1), or 6. That means that -1 is 6 units away from 5 on the number line. In each case, we knew the order of the subtraction, because we knew which was the bigger number.

But how do we deal with |x – 5|? Geometrically, this expression means “the distance from x to 5 on the number line. If x > 5, we would need to do x - 5 to find that distance. If x < 5, we would need to do 5 - x. Using |x - 5| handles both situations, because the order of the subtraction won’t matter; the absolute value symbol will ensure a nonnegative result. The meaning of something like |x - 5|, then, is the distance from x to 5 on the number line.

Example 1

Solve: |x| = 3

Solution 1

x = 3 or x = -3

Example 2

Solve:

|x - 6|=4

Solution 2

You can think about this in one of two ways:

a) Geometrically, this means “the distance from x to 6 on the number line is 4. If we go 4 units to the left of 6, we’ll land on 2. If we go 4 units to the right of 6, we’ll land on 10.

x = 2 or x = 10.

b) We can also look at this equation as the absolute value of a quantity (namely, x – 6) is equal to 4. That means that either x – 6 = 4 or x – 6 = -4. These two equations yield x = 10 or x = 2, same as our solution in part (a).

Example 3

Solve: |x - 3| < 5

Solution 3

This means geometrically, that we are looking for all values of x that are less than 5 units away from 3 on the number line. Going 5 units left of 3 gives us -2 and 5 units right of 3 gives us 8. That means that x has to be between -2 and 8.

The solution is { x : -2 < x < 8 }.

In interval notation, this solution is (-2, 8), where the parentheses indicate not including the numbers -2 and 8, but all numbers between them.

Example 4

Solve: |5 + x| ≥ 2

Solution 4

Let’s look at this in another way. If the absolute value of a quantity is greater than or equal to 2, then the quantity has to be greater than or equal to 2 or less than or equal to -2. This leads us to two inequalities:

5 + x ≥ 2 or 5 + x ≤ -2
x ≥ -3 or x ≤ -7

The solution is all values of x such that x ≥ -3 or x ≤ -7.

In interval notation, this looks like (-∞, -7] U [-3, ∞)

Note how the brackets indicate that we include the numbers -7 and -3 in our solution. We must use parentheses on infinity, because infinity is not actually a number, and therefore cannot be included in the solution.

Example 5

Joan’s physical trainer told her that her ideal weight is 125 pounds. She is currently within 5% of her ideal weight. What are the possible values of Joan’s current weight?

Solution 5

5% of Joan’s ideal weight is 5% of 125, or .05 • 125, or 6.25 pounds. Joan’s current weight could be modeled using absolute value. If we let W be Joan’s current weight, then | W – 125 | < 6.25.

-6.25 < W – 125 < 6.25
118.75 < W < 131.25

Joan’s current weight is between 118.75 pounds and 131.25 pounds.