Math Notes about Absolute Value, Absolute Equation, and Absolute Inequalities; This note is developed to engage readers while reading it.

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MATH GRADE 11 SOCIAL NOTES
Topic : Absolute Value, Absolute Equation, and Absolute Inequalities
Today’s Learning Outcomes :
1) Students understand the definition of absolute value
2) Students can determine the solution of absolute equation
3) Students understand the basic properties of absolute inequalities
4) Students can determine the solution of absolute inequalities
A. Absolute Value
Examples :
a) | 5 | = 5 d) | −2 | = 2
b) | 8 | = 8 e) | −9 | = 9
c)
7
2
7
2
= f)
8
5
8
5
=
−
d) 33 = g) 77 =−
By observing the examples above, the absolute value is …………………………………….
In math symbol and notation, the absolute value is notated by :



<−
≥
=
0,
0,
afora
afora
a
B. Absolute Equation
Example :
1) Determine the value of x which satisfies | 3x | = 12.
From the definition of absolute value, we get :
3x = 12 OR −(3x) = 12
x = 4 OR x = −4
The solution is { −4, 4 }
2) Determine the value of x which satisfies | 2x – 5 | = 3
2x – 5 = 3 OR −(2x – 5) = 3
2x = 8 OR 2x – 5 = −3
x = 4 OR 2x = 2
x = 1
The solution is { 1, 4 }
3) Determine the value of x which satisfies | x2
– x – 4 | = 2
x2
– x – 4 = 2 OR x2
– x – 4 = −2
x2
– x – 6 = 0 OR x2
– x – 2 = 0
(x + 2).(x – 3) = 0 OR (x – 2).(x + 1) = 0
x = −2 OR x = 3 OR x = 2 OR x = −1
1

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The solution is { −2, −1, 2, 3 }
4) Determine the value of x which satisfies 2
4
13
=
+
−
x
x
2
4
13
=
+
−
x
x
OR 2
4
13
=





+
−
−
x
x
02
4
13
=−
+
−
x
x
OR 2
4
13
−=
+
−
x
x
0
4
4
2
4
13
=





+
+
−
+
−
x
x
x
x
OR 0
4
4
2
4
13
=





+
+
+
+
−
x
x
x
x
0
4
9
=
+
−
x
x
OR 0
4
75
=
+
+
x
x
x = 9 ( why ?? ) OR
5
7−
=x ( why ?? )
The solution is {
5
7−
, 9 }
C. Some Properties of Absolute Inequalities
Examples :
1) | x | < 4 2) | x | > 2
x | x | x | x |
−5 −5
−4 −4
−3 −3
−2 −2
−1 −1
0 0
1 1
2 2
3 3
4 4
5 5
By observing the result in the table above,
1) | x | < 4 means ________________________
2) | x | > 2 means ________________________
Generally, it can be written :
Property #1. | x | < a ↔ ________________________
Property #2. | x | > a ↔ ________________________
D. Absolute Inequalities
2

3. 4
5
2
0 ∞
2
2
7
∞ 0
© fndnz sdhdm 1415
Examples :
1) Solve | 2x + 5 | ≤ 3.
Using Property #1, we get −3 ≤ 2x + 5 ≤ 3
−8 ≤ 2x ≤ −2
−4 ≤ x ≤ −1
Thus the solution is { x | −4 ≤ x ≤ −1 }
2) Solve | 3x – 4 | > 11
Using Property #2, we get 2x + 5 ≤ − 11 OR 2x + 5 ≥ 11
2x ≤ − 16 OR 2x ≥ 6
x ≤ − 8 OR x ≥ 3
Thus the solution is { x | x ≤ − 8 OR x ≥ 3 }
3) Solve 3
2
1
<
−
+
x
x
.
Using Property #1, we get 3
2
1
3 <
−
+
<−
x
x
.
Therefore,
2
1
3
−
+
<−
x
x
AND 3
2
1
<
−
+
x
x
0
2
1
2
)2(3
<





−
+
−
−
−−
x
x
x
x
AND 0
2
2
3
2
1
<





−
−
−
−
+
x
x
x
x
0
2
54
<
−
+−
x
x
AND 0
2
72
<
−
+−
x
x
AND
Combining the result above, we get :
Thus, the solution is { x |
2
7
4
5
>< xORx }
3
2
0 ∞ 0