Absolute value
Absolute value (second instance) is a Grade 7 math skill from Yoshiwara Intermediate Algebra reinforcing that |x| gives the non-negative distance of x from zero. Students apply this in equations, inequalities, and expressions, understanding that |x| equals x when x >= 0 and -x when x < 0.
Key Concepts
Property The absolute value of $x$ is defined by $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ x & \text{if } x < 0 \end{cases}$$ Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.
Examples To evaluate $| 9|$, since $ 9 < 0$, we use the second case of the definition: $| 9| = ( 9) = 9$.
To evaluate the expression $10 |5 8|$, first compute the operation inside the absolute value bars: $5 8 = 3$. Then take the absolute value: $| 3| = 3$. Finally, subtract: $10 3 = 7$.
Common Questions
What is the formal definition of absolute value?
|x| = x if x >= 0, and |x| = -x if x < 0. In both cases the result is non-negative.
How do you simplify |-8 + 3|?
First compute inside: -8 + 3 = -5. Then |-5| = 5.
How is absolute value used in distance?
|a - b| gives the distance between a and b on the number line. It is always non-negative.
What is the graph of y = |x|?
The graph of y = |x| is a V-shape with vertex at the origin, slope 1 for x > 0 and slope -1 for x < 0.