Absolute Value Equations
Hi, and welcome to this video about absolute value! We’ll cover what the absolute value of a number is and how to calculate it, as well as what absolute difference is and how it relates to absolute value. Let’s get started!
The absolute value of a number is its distance from 0 on a number line. Let’s use the number 6 as an example. Using a number line, we can see that 6 is 1, 2, 3, 4, 5, 6 units away from 0. Therefore, the absolute value of 6 is 6. Negative 6 is also 1, 2, 3, 4, 5, 6 units away from 0. The absolute value of – 6 is also 6.
Absolute value has a specific notation in mathematical expressions. For instance, the absolute value of 6 is denoted with two vertical lines here and here. And the absolute value of -6 is denoted as -6 with lines on either side, just as before.
Now, as we just saw, the absolute value of six equals six and the absolute value of -6 also equals 6.
No matter what number you are dealing with, big or large, positive or negative, the absolute value operation always yields a positive number. This is good to remember when evaluating expressions and solving equations, but it’s also important to remember why.
When we talk about distance, the direction doesn’t matter. In everyday language, we wouldn’t say, for instance, “The grocery store is negative 1 mile away.” We’d simply say “The grocery store is a mile away.”
Similarly, on the number line, we are simply counting distance without specifying direction. “So the absolute value of seven equals seven literally translates to -7 is 7 units away from 0.”
When we evaluate expressions and solve equations, the absolute value brackets act very much like parentheses do in the order of operations. We evaluate what’s inside the absolute value brackets first. Let’s look at a simple example.
We evaluate what’s inside the brackets first, so 2 plus -5 equals -3, which gives us 1 minus the absolute value of -3. Using our number line, we can see that -3 is three units away from zero, which means the absolute value of -3 is 3. This gives us 1 minus 3, which equals -2.
Since absolute value is a calculation of a number’s distance from zero, an absolute value expression such as the absolute value of 3 can be written as the absolute value of 3-0. Another way to interpret this statement is by saying “The distance between 3 and 0 is 3.” Writing it this way provides the same result. The second number here doesn’t need to be 0, though. We can actually calculate the distance between any two numbers.
On the number line, we can see the distance between -2 and 8 is 10. As an expression, we might write it as the absolute value of -2 minus 8 equals 10. This is called an absolute difference.
Let’s do a quick recap. The absolute value of a number is its distance from zero on a number line, and regardless of what number you’re working with, the answer will always be a positive number. The absolute difference is used to calculate the distance between any two numbers on the number line, not just a number and zero.
I hope this review was helpful! Thanks for watching, and happy studying!