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try #7 of absolute value notes on equations and inequalities. This is from the SMART Board, some formatting is missing. let me know of any mistakes.
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For those of you who missed it...
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| + 3 f(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 3 2x + 15 = 7x
15 = 5x 3 = x
2x + 12 = (7x 3) 2x + 12 = 7x + 3 9x + 12 = 3
9x = 9 x = 1
|18| = 18 |10| = 10
reject!
Absolute Value Inequalities Recall: |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 >5 written as
1 0 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
2x1<11 and 2x1>11 2x < 12 and 2x > 10 x < 6 and x > 5
1 0 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 ≥ 4 2x ≤ 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 < 5 written as
1 0 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
2x1>11 or 2x1<11 2x > 12 or 2x < 10 x > 6 or x < 5
1 0 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 ≥ 5 4|2x + 3| ≥ 16 |2x + 3| ≥ 4
2x + 3 ≥ 4 OR 2x + 3 ≤ 4 2x ≥ 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| ≤ 7
x + 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.
1 0 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
