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Following is Covered in the Video
Theory
What is Absolute Value / Modulus of a number Absolute Value on Number Line Properties of Absolute Values Absolute Value on Number Line Examples
What is Absolute Value / Modulus of a number
• Absolute Value or modulus (|x|) of a real number x is the non-negative value of the number (x), without any consideration to its sign Ex |12| = 12 |-12| = 12 (we just the value after ignoring the sign)
• |x| = x for x >0 = -x for x <0 = 0 for x = 0
Q1. Find the value of |-3| + | 2*3 – 4*2| + |25|
Q2. Find the value of | x+y| where x + z = 20 and y – z = -25
Sol1: 3 + | 6-8 | + 25 = 3 + 2 + 25 = 30
Sol2: x + z = 20 and y – z = -25 Adding both of them we get x + y = -5 => | x+y | = |-5| = 5
Absolute Value on Number Line
• Absolute value of a number x can also be imagined as the distance of that number x from 0 on a number line
Let's say we have two numbers x and y and x is positive and y is negative. What we are saying is |x| = x = distance of x from origin |y| = -y = distance of y from origin As, shown in the image below:
Properties of Absolute Values
• PROP 1: Absolute value of a number is always Non-negative |a| ≥ 0 for all values of a Ex: |3| = 3 ≥ 0 |-7| = 7 ≥ 0
• PROP 2: Minimum value of |a| = 0, when a=0 Ex: If |x| =0 => x=0
• PROP 3: Square root of a number is always positive \(\sqrt{a^2}\) = |?| Ex: If x = \(\sqrt{25}\) => x = +5 But if \(x^2\) = 25 => x = ± \(\sqrt{25}\) => x = ±5
• PROP 4: Absolute value of negative of a number is same as absolute value of the number |-a| = |a| A derivative of this is | a-b | = | b-a | because | b-a | = | -(a-b) |
• PROP 5: Product of absolute value of two numbers is same as product of their absolute values |ab| = |a|*|b| Ex: |7*3| = |7| * |3| = 21
• PROP 6: Division of absolute value of two numbers is same as division of their absolute values \(|\frac{?}{?}| = \frac{{|?|}}{{|?|} }\) Ex: \(|\frac{4}{2}| = \frac{{|4|}}{{|2|} }\) = 2
• PROP 7: Sum of absolute value of two numbers is always ≥ absolute value of their sum |a| + |b| ≥ |a+b| Ex: |7| + |3| ≥ |7+3| => 10 ≥ 10 |5| + |-8| ≥ |5 + (-8) | => 13 ≥ 3
• PROP 8: Difference of absolute value of two numbers is always ≤ absolute value of their difference |a| - |b| ≤ |a-b| Ex: |7| - |3| ≤ |7-3| => 4 ≤ 4 |5| - |-8| ≤ |5 - (-8) | => -3 ≤ 13
• PROP 9: Taking absolute value multiple times or taking it once gives the same result ||a|| = |a| Ex: ||-4|| = |-4| => |4| = |-4| = 4
• PROP 10: If absolute value of difference of two numbers is zero => both numbers are equal |a-b|=0 => a=b Ex: | x-4 | =0 > x=4
Next two will be used a lot to solve absolute values problem!
If a and b are two variables given then: |a-b| always means the distance between points a and b |a+b| = |a| + |b| when a and b have the same sign and |a+b| = |b| - |a| when a and b have different sign and |b| > |a|
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