18
The modulus of a number. Equations and inequalities with a modulus
Textbook: pp. 105–110
GoalKnow the definition and geometric meaning of the modulus; solve simple equations and inequalities with a modulus.
New words
modulus of a number · sonning modulidistance from 0 · 0 nuqtagacha masofaequation with a modulus · modulli tenglamainequality with a modulus · modulli tengsizlik
Explanation
The modulus of a number is written |a|: if a ≥ 0 then |a| = a, and if a < 0 then |a| = -a. So a modulus is never negative. Geometrically, |a| is the distance on the number line from the point a to the point 0. The equation |x| = a (a > 0) has two roots, x = a and x = -a; |x| = 0 only when x = 0; an equation like |x| = -3 has no solution. If an expression stands under the modulus, solve |f(x)| = a as f(x) = a or f(x) = -a. The inequality |x| < a (a > 0) is equivalent to -a < x < a, while |x| > a is equivalent to x < -a or x > a.
Worked examples
|2x - 1| = 5: 2x - 1 = 5 or 2x - 1 = -5; x = 3 or x = -2.
|x - 2| < 3: -3 < x - 2 < 3, add 2 to each part: -1 < x < 5. |x| ≥ 4: x ≤ -4 or x ≥ 4.
Class activity
“Distance game”: the class draws a number line on the floor; a pupil stands at a and counts steps to 0 — this gives |a|.
Practice
1
Compute |-7| + |3| - |-2|.
8
2
Solve |x + 4| = 0.
x = -4
3
Solve |x| ≥ 2.
x ≤ -2 or x ≥ 2
4
Why does |x| = -3 have no solution?
Because a modulus is never negative: the modulus of any number is at least 0.