Problems on Absolute Values

Mika Seppälä: Problems on Absolute Values

Equations

1 |2x – 8| = 2.

Solve the following equations:

|1 – |x|| = 3.2

|1 – x| + 2|x2 – 1| = 0.3

|2 – x| + |x2 – 4| = 4.4

Mika Seppälä: Problems on Absolute Values

Inequalities

5 |3x – 7| ≤ 2.

Solve the following inequalities:

|1 – x| + |x+1| ≤ 3.6

|2 –|x|| ≤ 17

Mika Seppälä: Problems on Absolute Values

Graphs

8 f(x) = |1 – |x – 1||, -1 ≤ x ≤ 3

Sketch the graphs of the following functions on the given intervals:

g(x) = |1 – |x – 3| + |x – 1|| , 0 ≤ x ≤ 49

h(x) = ||x2 – 4| – 5|, -4 ≤ x ≤ 410

Mika Seppälä: Problems on Absolute Values

Challenge Problems

11 For which value of the parameter r the equation

|x – 2| + |x – 4| = r

has infinitely many solutions? Interpret the problem geometrically. Find these solutions.

Show that x2 + 1 ≥ 2|x| for all x.12