IB MYP 4-5 Maths- Absolute values - Study Notes - New Syllabus
IB MYP 4-5 Maths- Absolute values – Study Notes
Standard
- Absolute values
IB MYP 4-5 Maths- Absolute values – Study Notes – All topics
Absolute Values
Absolute Values
The absolute value of a number is the distance of the number from 0 on the number line. Since distance is never negative, the absolute value is always zero or positive.
It is written using vertical bars: \( |a| \).
Examples:
- \(|5| = 5\)
- \(|-5| = 5\)
- \(|0| = 0\)
Visual on the Number Line:
Both \( 5 \) and \( -5 \) are 5 units away from 0, so their absolute values are the same:
Why It Matters (IB MYP Context):
In real life and MYP applications, absolute values are useful in:
- Calculating distances (e.g. how far two cities are apart on a map, regardless of direction)
- Banking (e.g. how much money you need to pay back, even if your balance is negative)
- Science experiments (e.g. difference in temperature readings)
Rules and Properties of Absolute Value:
| Property | Description |
|---|---|
| \( |a| = a \) | If \( a \geq 0 \) |
| \( |a| = -a \) | If \( a < 0 \) |
| \( |ab| = |a||b| \) | Product Rule |
| \( \left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|} \) | Quotient Rule ( \( b \ne 0 \) ) |
| \( |a – b| \) | Distance between numbers \( a \) and \( b \) |
| \( |a + b| \leq |a| + |b| \) | Triangle Inequality |
Solving Equations with Absolute Value:
To solve an equation like \( |x| = a \), where \( a \geq 0 \):
Important Note: An equation like \( |x| = -3 \) has no solution, because absolute value is never negative.
Example:
Evaluate the following expressions:
- \(|{-9}|\)
- \(|0|\)
- \(|3 – 7|\)
- \(|-5| + |2|\)
▶️ Answer/Explanation
Apply the definition of absolute value
\(|{-9}| = 9\)
\(|0| = 0\)
\(|3 – 7| = |-4| = 4\)
\(|-5| + |2| = 5 + 2 = 7\)
Example:
Solve the equation: \( |x| = 6 \)
▶️ Answer/Explanation
Consider both cases for the absolute value
\( x = 6 \) or \( x = -6 \)
Example:
Solve the equation: \( |x – 3| = 5 \)
▶️ Answer/Explanation
Consider both cases for the expression inside the absolute value
\( x – 3 = 5 \Rightarrow x = 8 \)
\( x – 3 = -5 \Rightarrow x = -2 \)
Example:
A submarine is at a depth of 300 m below sea level. A helicopter is flying at 450 m above sea level. What is the vertical distance between the submarine and the helicopter?
▶️ Answer/Explanation
Use the absolute value of the difference in positions
Submarine: \(-300\), Helicopter: \(+450\)
Distance = \( |-300 – 450| = |-750| = 750 \, \text{m} \)
Example:
Determine all values of \( x \) such that \( |2x + 1| = 7 \)
▶️ Answer/Explanation
Split into two cases
\(2x + 1 = 7 \Rightarrow 2x = 6 \Rightarrow x = 3\)
\(2x + 1 = -7 \Rightarrow 2x = -8 \Rightarrow x = -4\)
Example:
In a mountain region, a base camp is located at an altitude of 850 m above sea level. A cave is located at 420 m below sea level. A drone flies from the cave to the base camp.
Calculate the distance the drone travels.
▶️ Answer/Explanation
Represent the positions with signed numbers
Base camp: \( +850 \, \text{m} \)
Cave: \( -420 \, \text{m} \)
Use the absolute value of the difference
distance = \( |850 – (-420)| = |850 + 420| = |1270| \)
Simplify
\( |1270| = 1270 \, \text{m} \)