Absolute value
Evaluate absolute value expressions and solve absolute value equations: |x| measures distance from zero on the number line, always yielding a non-negative result regardless of sign.
Key Concepts
The absolute value of a number is its distance from the origin on the x axis. It's always non negative. If $x < 0$, then $|x| = x$. If $x \geq 0$, then $|x| = x$.
For $| 8|$, since $ 8 < 0$, the result is $ ( 8) = 8$. For $|12|$, since $12 0$, the result is simply $12$. For $|0|$, the result is $0$, as it has no distance from itself.
Think of absolute value as a 'positivity machine.' It doesn't care if you have 50 dollars or a debt of 50 dollars, it only sees the amount: 50! The distance from zero is always positive. This is why the absolute value of any non zero number, whether it's $ 10$ or $10$, always comes out as a positive value.
Common Questions
What does absolute value mean and how is it evaluated?
Absolute value |x| represents the distance from x to zero on the number line, which is always non-negative. If x>=0 then |x|=x. If x<0 then |x|=-x which is positive. For example |-7|=7 and |7|=7.
How do you solve an absolute value equation like |2x-3|=7?
An absolute value equation |expression|=k splits into two cases when k>0: the expression equals k or the expression equals -k. For |2x-3|=7, case 1 gives 2x-3=7 so x=5; case 2 gives 2x-3=-7 so x=-2. Check both solutions in the original equation.
When does an absolute value equation have no solution?
If the equation is set equal to a negative number like |x+1|=-3, there is no solution because absolute value is always non-negative. The left side can never equal a negative value, so the solution set is empty.