Download - Absolute Value Notes
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Absolute Value!!!
1. 2. 3.
4. 5.
6.
Domain: (-∞,∞)Range: [2,∞)
Domain: (-∞,∞)Range: (-∞, -2]
Domain: (-∞,∞)
Range: [-1,∞)
Domain: (-∞,∞)
Range: [-2,∞)
Domain: (-∞,∞)
Range: (-∞, 3]Domain: (-∞,∞)
Range: (-∞, 1]
Answers to Absolute Value Worksheet
f(x) = 2|x - 3| + 3f(x) = 1/3|x + 5| + 3
Answers to Absolute Value Worksheet
f(x) = -3/2|x - 5| - 4 f(x) = -3/2|x + 6| - 2
Answers to Absolute Value Worksheet
f(x) = 3|x - 4| - 10
f(x) = -2|x - 4| + 9
Answers to Absolute Value Worksheet
f(x) = 4|x + 5| + 9 f(x) = 3/5|x + 4| - 8
Answers to Absolute Value Worksheet
2x + 3 = 6 2x + 3 = -6 2x = 3 2x = -9 x = 3/2 x = -9/2
Solving Absolute Value Equations...Absolute Value:For any real number x,
|x| ={-x, if x < 0 0, if x = 0 x, if x > 0
Recall: When solving equations, isolate the absolute value. Here are a few examples...
1. 5|2x + 3| = 30 |2x + 3| = 6
Don't forget to check!!!
5|6| = 30 5|-6| = 30
solution set: {3/2, -9/2}
example 2: -2|x + 2| + 12 = 0
-2|x + 2| = -12 |x + 2| = 6
isolate the absolute value!
x + 2 = -6 x =
-8
x + 2 = 6 x =
4-2|-6| + 12 = 0 -2|6| + 12 = 0
{4, -8}
5|3×+ 7|=-65|3x + 7|=-13
absolute value cannot be negative!!
example 3:
{}
{3}
example 4:
|2x + 12| = 7x - 3
2x + 12 = 7x - 32x + 15 = 7x 15 = 5x 3 = x
2x + 12 = -(7x - 3)2x + 12 = -7x + 39x + 12 = 3
9x = -9 x = -1
|18| = 18|10| = -10
reject!
Absolute Value InequalitiesRecall: |ax+b|=c, where c>0
ax+b=c ax+b=-c|ax+b|<c think: between "and"
-c < ax+b < c
ax+b < c and ax+b > -c
ax+b>c or ax+b<-c
why?
we will express < or ≤ as an equivalent conjunction using the word AND
|ax+b|>c think: beyond "or" we will express > or ≥ as an equivalent disjunction using the word OR
I. Less than...a) |x| < 5
x < 5 and x >-5written as
10 2 3 4 5 6 7 8 9 10-1-2
-3-4-5-6-7-8-9-10
solution set: {x: -5< x < 5}
Graph on a number line!
use open circles!
shade between!!!
b) |2x - 1| < 11
2x-1<11 and 2x-1>-11 2x < 12 and 2x > -10 x < 6 and x > -5
10 2 3 4 5 6 7 8 9 10-1-2
-3-4-5-6-7-8-9-10
{x: -5 < x < 6}
c) 4|2x + 3| - 11 ≤ 5
4|2x + 3| ≤ 16 |2x + 3| ≤ 4
2x + 3 ≤ 4 AND 2x + 3 ≥ -42x ≤ 1 AND 2x ≥ -7 x ≤ 1/2 AND x ≥ -7/2
-1 0-2
-3-4-5 1 2 3 4 5
notice closed ends!
d) |7x + 10| < 0
think....can an absolute value be negative???NO!! {}
II. Greater than...a) |x| > 5
x > 5 or x < -5written as
10 2 3 4 5 6 7 8 9 10-1-2
-3-4-5-6-7-8-9-10
solution set: {x: x > 5 or x < -5}
Interval notation (we will not use this, just set, but as an FYI): (-∞, -5) ∪ (5, ∞)
Graph on a number line! use open circles!
shade beyond!!!
b) |2x - 1| > 11
2x-1>11 or 2x-1<-11 2x > 12 or 2x < -10 x > 6 or x < -5
10 2 3 4 5 6 7 8 9 10-1-2
-3-4-5-6-7-8-9-10
{x: x > 6 or x < -5}
c) 4|2x + 3| - 11 ≥ 54|2x + 3| ≥ 16 |2x + 3| ≥ 4
2x + 3 ≥ 4 OR 2x + 3 ≤ -42x ≥ 1 OR 2x ≤ -7 x ≥ 1/2 OR x ≤ -7/2
-1 0-2
-3-4-5 1 2 3 4 5
notice closed ends!
d) |7x + 10| > 0
think....when is an absolute value greater than 0???
always!!
{x: x ∈ R }x is a real number!
-1 0-2
-3-4-5 1 2 3 4 5
LAST ONE!
5 < |x + 3| ≤ 7
|x + 3| >5 |x + 3| ≤ 7x + 3 > 5 or x + 3 < -5 x+ 3 ≤ 7 and x + 3 ≥ -7 x > 2 or x < -8 x ≤ 4 and x ≥ -10
now graph it! graph above the number line and look for the overlap. This is where your solution will appear.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
{x: -10 ≤ x < 8 or 2 < x ≤ 4}
Remember to see me, email me or ask on the wiki if you have questions!!
-Ms. P