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Number - Absolute Value and Representation on the Number Line

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The absolute value of a real number xx, denoted by ∣x∣|x|, is its non-negative distance from zero on the number line, regardless of direction.

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For any real number xx, the absolute value is defined as: ∣x∣=x|x| = x if x≥0x \ge 0 and ∣x∣=−x|x| = -x if x<0x < 0.

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Absolute value is always non-negative: ∣x∣≥0|x| \ge 0 for all xx.

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On the number line, positive numbers are represented to the right of zero, and negative numbers are represented to the left of zero.

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The distance between any two points aa and bb on the number line is given by the expression ∣a−b∣|a - b|.

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Opposite numbers, such as 55 and −5-5, have the same absolute value because they are the same distance from zero: ∣5∣=∣−5∣=5|5| = |-5| = 5.

📐Formulae

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}

∣a∣=∣−a∣|a| = |-a|

Distance between a and b=∣a−b∣\text{Distance between } a \text{ and } b = |a - b|

x2=∣x∣\sqrt{x^2} = |x|

💡Examples

Problem 1:

Evaluate the expression: ∣−12∣+∣4∣−∣−7∣|-12| + |4| - |-7|

Solution:

12+4−7=912 + 4 - 7 = 9

Explanation:

First, find the absolute value of each term: ∣−12∣=12|-12| = 12, ∣4∣=4|4| = 4, and ∣−7∣=7|-7| = 7. Then perform the arithmetic: 12+4=1612 + 4 = 16, and 16−7=916 - 7 = 9.

Problem 2:

Find the distance between the points −8-8 and 55 on the number line.

Solution:

∣5−(−8)∣=∣5+8∣=∣13∣=13|5 - (-8)| = |5 + 8| = |13| = 13

Explanation:

The distance between two points aa and bb is ∣a−b∣|a - b|. Substituting the values, we get ∣5−(−8)∣|5 - (-8)|. Simplifying the signs gives ∣13∣|13|, which equals 1313 units.

Problem 3:

Solve for xx: ∣x−3∣=5|x - 3| = 5

Solution:

x−3=5 or x−3=−5x - 3 = 5 \text{ or } x - 3 = -5 x=8 or x=−2x = 8 \text{ or } x = -2

Explanation:

The equation ∣x−3∣=5|x - 3| = 5 means the distance between xx and 33 is 55 units. This can happen in two directions: x−3=5x - 3 = 5 (to the right) or x−3=−5x - 3 = -5 (to the left). Solving both linear equations gives x=8x = 8 and x=−2x = -2.