Absolute Value

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.

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 \(\left|-2\right|\) is equal to 2.

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

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

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.

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

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

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

$$\left|2x\right|=0$$

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.

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

\(\left|2x\right|+4=0\)

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

\(\left|any\;expression\right|+positive\;number=0\)

NEVER TRUE

NEVER TRUE

NEVER TRUE