Absolute Value

Hello, young mathematicians! Step into the world of absolute value, where numbers reveal their distance from zero, regardless of their positive or negative nature.

This lesson will explore the concept’s power, from understanding the notation |x| to its real-world applications.

Absolute value isn’t just about math; it’s a tool that empowers you to grasp magnitude in a whole new way. Let’s embark on this numerical journey together!

What Does Absolute Value Mean? Unveiling the Distance Within Numbers

Imagine you’re on a road trip and want to know how far you are from your starting point, regardless of whether you’re ahead or behind. That is the essence of absolute value. It is a way to measure a distance from zero on the number line without worrying about whether the number is positive or negative.

The absolute value of a number, denoted by |x| (with vertical bars), tells you how far that number is from zero. Whether it’s a positive number or a negative one, the absolute value is always positive or zero.

For instance, think about -5 and 5. They’re both 5 units away from zero, right? So, the absolute value of -5 is 5, and the absolute value of 5 is also 5.

Here’s the magic: |x| = x if x is positive or zero, and |x| = -x if x is negative. Absolute value strips away the negative sign and focuses solely on the distance, making it a versatile tool in math and beyond.

Absolute Value in Action: Examples and Equations

Let’s dive into some examples and equations that showcase the practical application of absolute value:

Example 1: Solving Equations Consider the equation |2x – 5| = 9. To solve for x, we need to find the values of x that make the absolute value equal to 9. It  means 2x – 5 could be either 9 or -9. Solve for both possibilities to find the solutions: 2x – 5 = 9 (x = 7) or 2x – 5 = -9 (x = -2).

Example 2: Distance Interpretation Imagine you’re standing at position -3 on a number line. How far are you from zero? Calculate the absolute value of -3: |-3| = 3. You’re 3 units away from zero.

Example 3: Real-Life Application Suppose you’re measuring temperatures. The absolute value of a negative temperature gives you the equivalent positive temperature. If it’s -10°C, the absolute value is 10°C – both temperatures are 10 degrees away from zero.

Example 4: Inequalities Let’s solve the inequality |x – 4| < 6. Break it into two cases:

1. x – 4 < 6 ⇒ x < 10
2. -(x – 4) < 6 ⇒ -x + 4 < 6 ⇒ -x < 2 ⇒ x > -2

Combine both solutions: -2 < x < 10.

Example 5: Practical Context Imagine you’re measuring the time difference between two events. If one happened 7 seconds ago and the other 15 seconds ago, what’s the absolute value of the time difference? It’s |15 – 7| = 8 seconds.

Practice problems

Problem 1: Equations Solve for x: a) |3x – 2| = 10 b) |2x + 1| = 5

Problem 2: Distances On a number line, if point A is at -6 and point B is at 8, find the distance between A and B.

Problem 3: Real-Life Application If the temperature outside is -5°C, what’s the absolute value temperature?

Problem 4: Inequalities Solve the inequality: |2y – 7| ≤ 3

Problem 5: Contextual Scenario You’re measuring the heights of two trees. Tree A is -12 feet tall, and Tree B is 20 feet tall. Calculate the absolute value of the difference in their heights.

ANSWERS:

Problem 1: a) x = 4 and x = -8/3 b) x = 2 and x = -3

Problem 2: The distance between A and B is 14 units.

Problem 3: The absolute value of the temperature is 5°C.

Problem 4: The solution to the inequality is 2 ≤ y ≤ 5.

Problem 5: The absolute value of the difference in tree heights is 32 feet.

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