Modulus Functions and Equations

SAT Questions that focus on Modulus Functions and Equations require knowledge of the following topics.

The Modulus Function

The Modulus Function is also referred to as Absolute Value Function, because it always returns the absolute value of an expression.

The modulus is expressed by two vertical bars, as in $y=|x-1|$.

For the sake of understanding, let's investigate the function $y=|x|$.
In this modulus function, when the value of "x" is a non-negative number, the function is "x"; otherwise "-x". For example,
for $x=5$, we have $y=|5|=5$; and
for $x=-5$, we have $y=|-5|=5$

Important Properties of the Modulus Function

P.1: $|x|\ge0$

P.2: $|x|\ge{x}$

P.3: $\sqrt{x^2}=|x|$

P.4: $|x.y|=|x|.|y|$

P.5: $|\frac{x}{y}|=\frac{|x|}{|y|}$

Graph of the Modulus Function

The graph of the modulus function is reflected in the x-axis.

Let's draw the graph of the function $f(x)=|x-3|$.
For $x\ge3$, we have $(x-3)\ge0$, therefore the modulus of the function is $(x-3)$.
For $x<3$, we have $(x-3)<0$, therefore the modulus of the function is $-(x-3)$.

The graph of this function will be reflected in the x-axis at point $x=3$, as shown in the following figure:

Solving Modulus Equations

Let'´s consider the following equation: $|2x-5|=11$.

The modulus of both 11 and -11 are equal to 11. Thus we can turn this modulus equation into two simple equations:
$2x-5=11$ and
$2x-5=-11$,

The value of “x” that satisfies the first equation is 8; and the value of “x” that satisfies the second equation -3. Therefore the values of "x" satisfying $|2x-5|=11$ are 8 and -3. Let's check it:

For $x=8$, we have $|2x-5|=|2*8-5|=|16-5|=|11|=11$
For $x=-3$, we have $|2x-5|=|2*(-3)-5|=|-6-5|=|-11|=11$

Solved SAT Practice Tests

Find some solved SAT Practice Tests in the following link:

SAT Modulus Functions and Equations