Absolute Value Inequalities
Q: How can we graph the solution of x  2 on a number line?
Since the x  2 yields the solution set {x: x = 2, -2}, we can conclude that
x  2 can be translated to -2 < x < 2.
Graph:
---------------------------------------------
Why?
ax  b  c can be translated into  c  ax  b  c
Process:
1. Remove the absolute value signs.
2. Set the expression between the positive and negative solution.
3. Solve the inequality
4. Graph the inequality.
**IMPORTANT TO KNOW**
ax  b  c translates to ax  b  c OR ax  b  c
Therefore, we may conclude that < gives us a compound inequality, using the term AND.
> gives us a compound inequality, using the term OR.
Ex:
Ex:
x4 3
3  x 4  3
+4
+4 +4
1 x  7
x 1  2
x  1  2
-1 -1
x  3
Ex:
---------------------------------------------
or
x 1  2
-1 -1
x 1
---------------------------------------------
Express 2  x  8 as an absolute value inequality.
---------------------------------------------
Process:
1. Locate the midpoint of the inequality graph.
2. Subtract that number from both sides of the inequality.
3. Express as an absolute value inequality.
2 x 8
-5 -5 -5
-3 < x – 5 < 3
~>
x 5  3
Examples: Solve and Graph the following.
1 3
1. x  1  3
2. x  
2 2
5 3

2 2
3. x  1  2
4. x  2  4
5. x 
6. x  2  1
7. x  2  5
8. 1  x  4
9. 2 x  1  3
10. 4  x  9
11. 4  x  2
12.  x  1  1
Homework: Work Book p.73, 16-27
Good Mixed Review: Work Book p. 74, ALL
***EXTRA PROBLEMS***
1. x  8  9
2. 9  x  7
3. 4  x  5
4. x  12  36
5. x  12  6
6. 5  x  18
7. x  1  17
8. x  9  4
9. 3x  6  0
10. 10  4 x  2
11. 1  2 x  9
12. 2 x  3  4