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```Warm-Up
• Evaluate each expression, given that x=3 and y=-2.
a. |2x -9|
1) -3
2) 3
3) 15
4) -15
2) 1
3) -1
4) 5
2) x= -1, 5
3) x= 3, -15
4) x= -3, 15
b. |y –x|
1) -5
• Solve.
|3x + 6| = 9
1) x=1, -5
6.4: Absolute Values and
Inequalities
Objective:
• Learn how to solve absolute
value inequalities.
Review
• Why is the absolute value of a number
always greater than or equal to zero?
• Two or more inequalities connected by
the words _______ or _________ are a
compound inequality.
Conjunction: |ax + b| < c
Means: x is between + c
-c < ax +b < c
Less Than when an absolute value is on
the left and the inequality symbol is < or
≤, the compound sentence uses and.
Disjunction:
Means:
|ax +b| > c
not between!
ax + b < -c or
ax + b > c
Greater Than when an absolute value is
on the left and the inequality symbol is >
or ≥, the compound sentence uses or.
Solving absolute inequalities
and graphing:
|x - 4| < 3
Means:
(less than is between)
-3 < x- 4 < 3 (solve)
+4
+4 +4
1< x< 7
Graph:
0 1 2 3 4 5 6 7 8 9
Solving absolute inequalities
and graphing:
• |s – 3| ≤ 12
(less than is between)
Means: -12 ≤ s – 3 ≤ 12 (solve)
+3
+3 +3
- 9 ≤ s ≤ 15
Graph:
-9 -6 -3 0 3 6 9 12 15 18 21 24
• Solve each absolute value inequalities
then graph.
• A. |y + 4| < 5
• B. |z – 3| ≤ 2
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
• Solve each absolute value inequalities
and graph.
• A. | 3y – 3| > 9
• B. |2x + 7| ≥ 11
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
|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
x+1 > 5
(write x + 1 inside the absolute
brackets and 5 outside positive)
Write an absolute value inequality for the
graph shown
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Closing the lesson:
• Summarize the major points of the