Distance and Absolute Value
Distance and absolute value is a Grade 7 math skill from Yoshiwara Intermediate Algebra linking the concept of absolute value |a - b| to the distance between two numbers on the number line. Students learn that |x - c| = d means x is exactly d units away from c.
Key Concepts
Property The distance between two points $x$ and $a$ is given by $|x a|$.
Examples The statement "$x$ is five units from the origin" can be written using absolute value notation as $|x| = 5$. The solutions are $x=5$ and $x= 5$. The statement "$p$ is two units from 7" can be written as $|p 7| = 2$. The solutions are $p=5$ and $p=9$. The statement "$a$ is within four units of $ 3$" can be written as $|a ( 3)| < 4$, which simplifies to $|a + 3| < 4$. This means $a$ is between $ 7$ and $1$.
Explanation Absolute value is a tool for measuring distance on a number line without worrying about direction. The expression $|x a|$ asks the question, "How far apart are $x$ and $a$?" The result is always a positive number.
Common Questions
How does absolute value relate to distance?
The absolute value |a - b| represents the distance between a and b on the number line. Distance is always non-negative.
What does |x - 3| = 5 mean geometrically?
It means x is exactly 5 units from 3 on the number line, giving two solutions: x = 8 and x = -2.
How do you solve |x - c| = d?
Set x - c = d and x - c = -d, giving x = c + d and x = c - d. These are the two points at distance d from c.
What is the absolute value of a negative number?
The absolute value of any number is its distance from zero, always non-negative. |−7| = 7.