Download 6.4: Connections: Absolute Values and Inequalities

6.4: Absolute
Values and
Inequalities
Conjunction: |ax + b| < c
Means: x is between + c
-c < ax +b < c
Disjunction:
Means:
|ax +b| > c
not between!
ax + b < -c or
ax + b > c
Solving absolute inequalities
and graphing:
|x - 4| < 3
Means:
(less than is betweeness)
-3 < x- 4 < 3 (solve)
+4
+4 +4
1< x< 7
Graph:
0 1 2 3 4 5 6 7 8 9
Solve and graph:
|x + 9 |> 13
(disjunction)
Means: x + 9 < -13
-9
-9
or
x + 9 > 13
x < -22
Graph:
-25 -20 -15 -10 -5 0 5 10
-9
-9
x>4
Change the graph to an absolute value
inequality:
0 1 2 3 4 5 6 7 8 9 10
1. Write the inequality. (x is between)
2<x<8
2. Find half way between 2 and 8
It’s 5 (this is the median)
To find the median, add the two numbers
and then divide by 2.
2+8 = 5
2
3. Now rewrite the inequality and
subtract 5 (the median) from each section.
2-5<x-5<8-5
Combine like terms or numbers and you
get -3 < x - 5 < 3
4. Write your absolute inequality
|x - 5| < 3
Notice: The median is 3 units away
from either number.
Write the inequality for this disjunction:
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
1. x < -6
+1 +1
2. x + 1 < - 5
3. |x+1|>5
or
x>4
+1 +1
(find the median)
(subtract -1 from both
sides, so add 1)
x+1 > 5
(write x + 1 inside the absolute
brackets and 5 outside positive)
Quick rule:
|x - median| ( inequality symbol here) range 2
Median:
add the two numbers together and divide
by 2. Remember to subtract. Watch signs!
Range:
subtract the two numbers, then divide by 2.