New Zealand
Level 7 - NCEA Level 2

# Simplify Expressions with Absolute Values

Lesson

There are several ways in which we can characterise the absolute value function.

Absolute value is used when we are concerned with the magnitude of a quantity and not with its sign. Thus, we can define the absolute value of a number to be the positive number that has the same magnitude as the number. The treatment of absolute values with numbers is described here, check that out to refresh.

The notation $|x|$|x| is used for the absolute value of $x$x.

For example, $|3|=3$|3|=3 and $|-5|=5$|5|=5.

We can define the function piecewise, as follows.

Alternatively, we can write $|f(x)|=\sqrt{\left(f(x)\right)^2}$|f(x)|=(f(x))2. This works because the square root sign is always understood to give the positive square root.

#### Let's look at some simple cases,

##### Case 1

Evaluate $\left|8-3\right|$|83|.

##### Case 2

The expression $\left|2x\right|$|2x| can be simplified for values of $x$x less than $0$0, or greater than $0$0

So, the expression is equivalent to $2x$2x for $x\ge0$x0 and $-2x$2x for $x<0$x<0

##### Case 3

Evaluate $\left|6-3\right|-\left|3-8\right|$|63||38|.

##### Case 4

Consider the expression $\left|x-y\right|-\left|y-x\right|$|xy||yx|.

1. Simplify the expression in the case where $x\ge y$xy.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems