1. Solve Absolute Value Inequalities

2. Conjunction: |ax + b| < c
*notice it is <c
Means: x is between + c
Rewrite the conjunction as
-c < ax +b < c
This is it’s equivalent compound inequality

3. Disjunction: |ax +b| > c *notice it is >c Means: not between!
Rewrite the disjunction as
ax + b < -c or ax + b > c
This is it’s equivalent compound inequality

4. *TIP to help you remember
Conjunctions
Has a < less than symbol and it requires less work
|2x - 1| < 6
Isolate the absolute value, then just drop the bars and put a –6 on the left of the compound inequality and a +6 on the right
-6 < 2x - 1 < 6
Then solve like normal
Disjunctions
Has a > greater than symbol and it requires a greater amount of work
|2x - 1| > 6
You must write 2 separate inequalities. Isolate the absolute value. Then for one, just drop the bars and use >6. For the other, drop the bars, flip the inequality sign and use <-6
2x - 1 < -6 or 2x - 1 >6
Then solve like normal

5. Solving absolute inequalities and graphing:
|x - 4| < 3 (less than is betweeness)
Means: -3 < x- 4 < 3
Graph:
(solve) +4 +4 +4
1< x< 7

6. Solve and graph: *you have to isolate the absolute value bars first!
|x + 3| -2 < -1
Means: -1 < x+3 < *now solve
Graph: +2 +2
|x + 3| < 1
- 3
- 3
- 3
-4 < x < -2

7. |x + 9 |> 13 (disjunction) Means: x + 9 < -13 or x + 9 > 13
Solve and graph:
|x + 9 |> 13 (disjunction)
Means: x + 9 < or x + 9 > 13 -9 -9 -9 -9
x < -22
x > 4
Graph:

8. To write an absolute value inequality from a graph (for Conjunctions)
1. Write the compound inequality of the graph. (x is between)
2 < x < 8
Find the median (half way between 2 and 8)
It’s 5
To find the median, add the two numbers and then divide by = 5 2

9. 3. Now rewrite the inequality and subtract the median (5) from each section.
2 - 5 < x - 5 < 8 - 5
Combine like terms (simplify the numbers) and you get -3 < x - 5 < 3
4. Now turn that into it’s corresponding conjunction absolute value inequality |x - 5| < 3
Notice: The median is 5, and the median is 3 units away from either ending number on the number line.

10. To write an absolute value inequality from a graph (for disjunctions)
Write the compound inequality of the graph.
Find the median. (it’s -1)
Subtract the median from all sides
(subtract -1 will be adding 1)
4. Write as abs.value inequality. Put x + 1 inside the absolute value bars and positive 5 on the other side of >
x < -6 or x > 4
x < -6 or x > 4 +1
x + 1 < or x+1 > 5
|x+1|>5

11. Quick rule: Median: Range:
|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.