Definition of Absolute Value
Consider the number line and a point at a distance x from the origin zero of the number line.
The absolute value of x written as x is the distance from zero to x. Since x gives a distance, it is always positive or equal to zero. Hence
 5  = 5
 5  = 5
 0  = 0
a  b is the distance between points at a and b on the number line.
5  15 =   10 = 10
definition
\[ x = \left\{
\begin{array}{ll}
x & x\geq 0 \\
x & x\lt 0 \\
\end{array}
\right. \]
which can be used to write
\[ x  a = \left\{
\begin{array}{ll}
x  a & x\geq a \\
 (x  a) = a  x & x\lt a \\
\end{array}
\right. \]
Absolute Value and Square Root
\( \sqrt { x^2} =  x  \) ; because the square root is positive or equal zero.
\( \sqrt {(x  a)^2} = x  a \)
Properties of the Absolute Value Function
 \(  x  \ge 0 \)
 \( \sqrt{ x^2} =  x  \)
 \(  x  =   x  \)
 \(  a b  =  a  b  \)
 \( \left  \dfrac{a}{b} \right  = \dfrac{a}{b} , b \ne 0\)
Equations and Inequalities with Absolute Value
For \( k \ge 0 \)
 \(  x  = k \iff x = k \text{ or } x =  k \)
 \(  x  \le k \iff  k \le x \le k \)
 \(  x  ≥ k \iff x \le  k \text{ or } x \ge k \)
Graph of Absolute Value of a Function
Function f(x) used is a quadratic function of the form
\[ f(x) = a x^2 + b x + c \]
The exploration is carried out by changing the parameters a, b and c included in \( f(x)\) above.
Interactive Tutorial
 click on the button above "draw" to start.
 Use the sliders to set parameter a to zero, parameter b to zero and parameter c to a positive value; \( f(x)\) is a constant function. Compare the graph of f(x) in blue and that of \( h(x)= f(x) \) in red. Change c to a negative value and compare the graphs again. Use the definition of the absolute value functions to explain how can the graph of \( f(x) \) be obtained from the graph of \( f(x)\).

Keep the value of a equal to zero, select non zero values for b to obtain a linear function . How can the graph of \( h(x) = f(x)\) be obtained from that of \( f(x)\)?
Hint: use the definition of the absolute value functions and reflection of a graph on the xaxis.
 Set b and c to zero and select a positive value for a to obtain a quadratic function . Why are the two graphs the same? (Hint: use the definition of the absolute value functions).
 Set b and c to zero and select a negative value for a to obtain a quadratic function . Why are the two graphs reflection of each other? (Hint: use the definition of the absolute value functions and reflection of a graph on the xaxis).
 Keep the values of a and b as in 5 above and change gradually c from zero to some positive values. How can the graph of \( h(x)=f(x)\) be obtained from that of \( f(x)\)?
 Select different values for a, b and c and explore.
ExercisesPart 1
Evaluate the expressions
   9 
  9 
  2  6 
 \(  \dfrac{5}{2} \)
 \(  11  4^2  \)
 \( 2^{ 2  7} \)
 \( 219  9  20 \)
Part 2
Substitute and Evaluate the expressions
  x  10  +  2 x  , x =  7
   x  , x = 8
 \( \left  \dfrac{x  2}{x + 9} \right  \) , x = 20
 \(   3 x  x^2  \) , x = 3
 \( x^{ x  7} \) , x =  2
Part 3
Simplify the expressions
  a  b  for a = b
  a  b  for a > b
  a  b  for b > a
 \( \sqrt {3^2} \)
 \( \sqrt {( 3)^2} \)
 \( \sqrt { (12  20)^2 } \)
 \(  x^2  \)
 \(  (x)^2  \)
 \(  (x + 1)^2  \)
 \( \sqrt{(x  y + z)^2} \)
 \( \sqrt{sin^2(x)} \)
Part 4
Solve the Equations
 \( x  2 = 4 \)
 \( x + 5 = 0 \)
 \( 10  x =  5 \)
Part 5
Solve the Inequalities
 \( x  7 \gt 4 \)
 \( x + 5 \lt 9 \)
 \( x + 8 \lt  5 \)
 \( x + 8 \gt  2 \)
Part 6
Sketch the following pairs of functions in the same system of axes and explains the similarities and differences of the two graphs.
 \( f(x) = x  1 , h(x) = f(x) \)
 \( f(x) = x^2  4 , h(x) = f(x) \)
 \( f(x) = \sin(x) , h(x) = \sin(x) \)
Solutions to the Above Exercises
Part 1
Evaluate the expressions
   9  = 9
  9  = 9
  2  6  = 4
 \(  \dfrac{5}{2} = \dfrac{5}{2} \)
 \(  11  4^2  = 5 \)
 \( 2^{ 2  7} = 32 \)
 \( 219  9  20 = 6 \)
Part 2
Substitute and Evaluate the expressions
 For x =  7,  x  10  +  2 x  = 31
 For x = 8 ,   x  = 8
 For x =  20 , \( \left  \dfrac{x  2}{x + 9} \right  = \dfrac{22}{29}\)
 For x =  3 , \(   3 x  x^2  = 0 \)
 For x =  2 , \( x^{ x  7} =  512\)
Part 3
Simplify the expressions
  a  b  = 0
  a  b  = a  b
  a  b  = b  a
 \( \sqrt {3^2} = 3 = 3 \)
 \( \sqrt {( 3)^2} = 3 = 3 \)
 \( \sqrt { (12  20)^2 = 8} \)
 \(  x^2  = x^2\)
 \(  (x)^2  = x^2\)
 \(  (x + 1)^2  = (x + 1)^2 \)
 \( \sqrt{(x  y + z)^2} = x  y + z\)
 \( \sqrt{sin^2(x)} = sin(x) \)
Part 4
Solve the Equations
 \( x  2 = 4 \) , two solutions: x =  2 and x = 6
 \( x + 5 = 0 \) , one solution: x =  5
 \( 10  x =  5 \) , no real solutions.
Part 5
Solve the Inequalities
 \( x  7 \gt 4 \) , solution set: x < 3 ∪ x > 11
 \( x + 5 \lt 9 \) , solution set: 14 < x < 4
 \( x + 8 \lt  5 \) , solution set: {∅}
 \( x + 8 \gt  2 \) , solution set: {all real numbers}
Part 6

\( f(x) = x  1 , h(x) = f(x) = x  1 \)
For x  1 ≥ 0 or x ≥ 1, h(x) =  x  1  = x  1 and we therefore have h(x) = f(x) which we can be clearly seen from the graphs.
For x  1 < 0 or x < 1 , h(x) = x  1 =  (x  1) =  f(x).
Therefore for x < 1 h(x) =  f(x) which means that the graph of h is a reflection of the graph of f on the x axis and this is can be clearly seen on the graphs.

\( f(x) = x^2  4 , h(x) = f(x) = x^2  4 \)
For x^{2}  4 ≤ 0 or 2 ≤ x ≤ 2 , h(x) = (x^{2}  4) =  (x^{2}  4) =  f(x) . Therefore, on the interval 2 ≤ x ≤ 2, the graph of h is a reflection of the graph of f. Outside this interval, the two graphs coincide.

\( f(x) = \sin(x) , h(x) = \sin(x) \)
Over intervals 2π to π , 0 to π , 2π to 3π, ... sin(x) ≥ 0 and h(x) = sin(x) = f(x) therefore the graphs of h and f coincide.
For intervals  π to 0 , π to 2π ... sin(x) < 0 , h(x) =  sin(x)  =  sin(x) and the graph of h is a reflection of the graph of f on the x axis.
Conclusion: In all three exercises above, the graph of h is on or above the x axis because h(x) = f(x) and the absolute value is positive or equal to 0.
More References and LinksSimplify Absolute Value Expressions.
Absolute Value Equations And Inequalities.
Absolute Value.
