Unit 54: Absolute Value

This section will take you through the basics of understanding absolute value for the Digital SAT.

Number lines are also useful for thinking about absolute value.  Absolute value is how far away from 0 the number is, whether going left (negative) or right (positive).  Thus,

In our example above, both -2 and 2 are two units from 0.  Both have an absolute value of 2.

A couple rules to remember when dealing with absolute values:

1) Absolute values can be 0, but they can never be negative.

Absolute value is represented mathematically by the symbol (left|right|) .  So, we would say (left|-2right|) is equal to 2.

Expressions can also be given within an absolute value.  For instance.

$$left|2x+3right|=3$$

To solve, we need to remember that whatever is in the (left|right|) can be equal to 3 OR -3.  From this, we can generate two equations.

$$2x+3=3\2x+3=-3$$

Solving each independently, we can determine that x can equal 0 or -3.

2) You CANNOT perform operations inside and outside of an absolute value.  For instance, with an equation

$$left|2x+3right|=3$$

We can’t just subtract 3 from both sides to get

$$left|2xright|=0$$

This will NOT lead us to two correct answers.  Don’t do it. The SAT knows students make this mistake all the time and will put an answer that looks right based on this common error.

3) Absolute Value boundaries can act as grouping symbols

Absolute value functions can sometimes act similarly to parentheses due to their property of creating a “boundary” around the expression within them. This rule is essential for when you are solving equations that combine like terms. For example in the equation

$$5left|3x-15right|+2left|3x-15right|=35$$

We would treat the absolute value boundaries as grouping symbols, much like parentheses, ensuring that the entire expression within the absolute value is treated as a single unit.

After combining like terms,

$$7left|3x-15right|=35$$

From here we would divide both sides of the equation by 7, and then split the equation into two and solve for both values of x.

One last rule – because absolute values cannot be negative, they can never equal a negative number or equivalent.

(left|2xright|+4=0)

(left|x+3right|=-1)

(left|any;expressionright|+positive;number=0)

NEVER TRUE

NEVER TRUE

NEVER TRUE

Strategy:

  1. Isolate the absolute value.
  2. Create two separate equations and solve.
    • Set up two separate equations to solve for each case: one where the absolute value is equal to the positive value of the other side, and one where it is equal to the negative value of the other side.