1. Problems on Absolute Values

2. Mika Seppälä: Problems on Absolute Values
Equations
Solve the following equations: 1
|2x – 8| = 2. 2
|1 – |x|| = 3. 3
|1 – x| + 2|x2 – 1| = 0. 4
|2 – x| + |x2 – 4| = 4.
Mika Seppälä: Problems on Absolute Values

3. Mika Seppälä: Problems on Absolute Values
Inequalities
Solve the following inequalities: 5
|3x – 7| ≤ 2. 6
|1 – x| + |x+1| ≤ 3. 7
|2 –|x|| ≤ 1
Mika Seppälä: Problems on Absolute Values

4. Mika Seppälä: Problems on Absolute Values
Graphs
Sketch the graphs of the following functions on the given intervals: 8
f(x) = |1 – |x – 1||, -1 ≤ x ≤ 3 9
g(x) = |1 – |x – 3| + |x – 1|| , 0 ≤ x ≤ 4 10
h(x) = ||x2 – 4| – 5|, -4 ≤ x ≤ 4
Mika Seppälä: Problems on Absolute Values

5. 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. 12
Show that x2 + 1 ≥ 2|x| for all x.
Mika Seppälä: Problems on Absolute Values