Download Transformations That Produce Equivalent Inequalities

ABSOLUTE VALUE
a = a if a ≥ 0
a = −a if a < 0
1. If a < b, then -b < a < b
2. If a > b, then -b > a > b
3. If a = b , then a = ± b
Open Interval:
a<x<b
denoted: (a,b)
Closed Interval:
a≤ x≤b
denoted: [a,b]
Half-Opened interval:
a≤ x<b
denoted: [a,b)
On a number line, the distance between the graphs of two numbers a and b
( |a – b| or |b – a|)
is the absolute value of the difference of a and b.
Example #2: Solve for x in the equality |x - 2| = 5
(Equivalent to the disjunction: x – 2 = 5 or x – 2 = -5)
x= 7 or x = -3
Example #3: Solve for x in the inequality |x + 5 | < 6 (Note: |x – (-5)| < 6
(Equivalent to the conjunction: -6 < x + 5 and x + 5 < 6).
Note: Less Thand.
-11 < x < 1 or x ∈ (−11,1)
Example 4: Solve for x in the inequality | 2 x − 3 |≥ 6
(Equivalent to the disjunction: 2 x − 3 ≤ −6 or 2 x − 3 ≥ 6 )
Note: Greator
2 x − 3 ≤ −6 or 2 x − 3 ≥ 6
2 x ≤ −3 or 2x ≥ 9
−3
9
x≤
or x ≥
2
2