Distance between points and equations with a modulus
Find the distance between two points of the number line as |a - b| and solve equations |x - a| = b by marking points.
- Palestine: grade 11
- Russia: grade 6
Guide
In the Russian programme (Grade 6) the modulus of a number is its distance from the origin on the number line. Then the distance between points a and b is |a - b|. Equations with a modulus are taught in Grades 6-7: |x| = a, |x - a| = b and |ax + b| = c.
Geometric meaning: solving |x - 2| = 3 means finding all points at distance 3 from the point 2. There are two: -1 and 5. If the right side is negative there is no solution, because a distance is never negative; if it is zero there is one solution.
For the linear equation |2x - 1| = 5 the expression inside is either 5 or -5, giving two equations. The inequality |x - a| < b is the interval (a - b; a + b), as an extra.