# Algebra: Understanding Absolute Value

The concept of absolute value is quite simple and is defined by the following:

In simple terms, the absolute value of a real number is non negative. This means |*x*| = *x* and |-*x*| = *x*. Therefore |4| = 4 and the |-4| = 4. The reason why absolute value is always positive is because it has to do with distance and distance cannot be negative, that is, it doesn’t matter how far you travel or in what direction, you cannot travel – 5 km or – 25 miles. Look at it this way: if you say what is the difference between your age and my age the answer cannot be a negative age. However, as it relates to absolute value it carries the idea of distance on the number line. So if we have -7, it is 7 places away from 0 and that 7 places is considered as positive taking into consideration what was explained before.

**Example 1**

What is |5 – 8|?

A common mistake that some persons make is they think that the 8 becomes positive (+) and therefore you have |5 + 8|. That is not the case. What you do is work the operation first just as if it was like (5 – 8), which will be (-3) or just -3. Therefore we should have |-3|. Since the absolute value of any real number is non negative then the final answer is 3.

**Example 2**

What is |(-4 + 5) – 9| + |8 + (-2 – 3)|?

Just as was explained above, you work out what is inside the modulus sign first. Modulus is another name for absolute value. Therefore:

|(-4 + 5) – 9| + |8 + (-2 – 3)| = |(1) – 9| + |8 + (-5)|

= |-8| + |3|

= 8 + 3

= 11.

The rules of absolute value is a bit different when we consider linear equations and inequalities. As it relates to the linear equations and inequalities we have to bear these in mind.

|A| = B => A = B or A = – B

|A| ≤ B => -B ≤ A ≤ B

|A| ≥ B => A ≥ B or A ≤ – B

At first glance you might not understand the rules but after we will look at them in more practical ways you will better appreciate them.

**Example 3**

Solve |*x* – 1| = 4.

When you look at the example, you may notice that it satisfies |A| = B. As such we would have to work it two ways:

*x*– 1 = 4*x*– 1 = – 4

Solving the first equation:

*x *– 1 + 1 = 4 + 1

*x *= 5.

Second equation:

*x – *1 = – 4

*x* – 1 + 1 = – 4 + 1

*x* = – 3

Therefore the answer is – 3 and 5. Which means that these are the values that satisfies the original problem |*x* – 1| = 4. We can check it just to see how it makes sense. If we substitute – 3 or 5 for *x *into |*x* – 1|, we should get 4. Let us try it:

When *x* = 5.

|5 – 1|

|4| = 4

When *x* = – 3

|-3 – 1|

|- 4| = 4. We note that the absolute value for any real number is non negative.

**Example 4**

Solve |2*x* + 3| < 15.

When we look at this, we note that it satisfies |A| ≤ B. Though the rule has the less than or equal to sign (≤), whatever sign we get, we use it. So by following the rule we should have:

-15 < 2*x* + 3 < 15

The above is what is called a *compound inequality*. Basically what you will do is actually solve two inequality simultaneously. The two inequalities are:

- 2
*x*+ 3 < 15 - 2
*x*+ 3 > – 15

If we were to solve the first inequality, we would have to subtract 3 from both sides then follow through and the same thing would have to be done to the other equation.

-15 – 3 < 2*x* + 3 – 3 < 15 – 3 = -18 < 2*x* < 12

-9 < *x* < 6

Therefore, all the values from – 8 to 5 will satisfy |2*x* + 3| < 15.

**Example 4**

|2*x* – 1| ≥ 5

The above example satisfies |A| ≥ B. Therefore based on the rule, we should have:

2*x* – 1 ≥ 5 or 2*x* – 1 ≤ – 5

So let us solve both equations:

2*x* – 1 + 1 ≥ 5 + 1

*x* ≥ 3

or

2*x* – 1 + 1 ≤ – 5 + 1

*x* ≤ – 2

Therefore, all the values that satisfy|2*x* – 1| ≥ 5, lie between the interval, – 2 ≥ *x *≥ 3. Which means *x* has to be a number less than or equal -2 and a number that is greater than or equal to 3.