What is an Equation with Absolute Values?

S3-SA1-0674

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

An equation with absolute values is a mathematical statement where a variable is inside the absolute value symbol ( | | ). The absolute value of a number is its distance from zero on the number line, always making it positive. So, these equations ask us to find numbers whose distance from zero is a specific value.

Simple Example

Quick Example

Imagine you are standing at '0' on a number line. If someone says, 'I am at a point whose distance from 0 is 5 steps,' you could be at +5 or -5. Both |+5| and |-5| equal 5. So, an equation like |x| = 5 means x can be 5 or -5.

Worked Example

Step-by-Step

Let's solve the equation |x - 3| = 7.

Step 1: Understand that the expression inside the absolute value, (x - 3), can be either 7 or -7, because both 7 and -7 have an absolute value of 7.

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Step 2: Set up two separate equations based on this understanding.
Equation 1: x - 3 = 7
Equation 2: x - 3 = -7

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Step 3: Solve Equation 1 for x.
x - 3 = 7
x = 7 + 3
x = 10

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Step 4: Solve Equation 2 for x.
x - 3 = -7
x = -7 + 3
x = -4

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Step 5: Check your solutions by plugging them back into the original equation.
For x = 10: |10 - 3| = |7| = 7. (Correct!)
For x = -4: |-4 - 3| = |-7| = 7. (Correct!)

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Answer: The solutions are x = 10 and x = -4.

Why It Matters

Understanding absolute value equations is crucial for fields like Physics to calculate distances or errors, and in Computer Science for algorithms that measure differences. Engineers use them to define tolerances for measurements, ensuring machines work correctly. Even in data science, they help measure how far data points are from an average.

Common Mistakes

MISTAKE: Distributing the absolute value (e.g., thinking |x - 3| is the same as |x| - |3|). | CORRECTION: The absolute value sign acts like a grouping symbol. You must evaluate the expression inside first, then take its absolute value. |x - 3| is not equal to |x| - |3|.

Practice Questions

Try It Yourself

QUESTION: Solve |y| = 9 | ANSWER: y = 9, y = -9

QUESTION: Find the values of 'a' that satisfy |a + 2| = 6 | ANSWER: a = 4, a = -8

QUESTION: Solve for 'z': |2z - 5| = 13 | ANSWER: z = 9, z = -4

MCQ

Quick Quiz

Which of the following equations has NO solution?

|x - 1| = 0

|x + 2| = 7

|3x| = -4

|x - 5| = 1

The Correct Answer Is:

An absolute value always represents a distance, which cannot be negative. Therefore, |3x| = -4 has no solution because an absolute value cannot equal a negative number.

Real World Connection

In the Real World

Imagine a drone delivering packages for a company like Zepto. If the drone's target delivery location is '0' on a map, and its GPS system says it's at a position 'x' such that |x| = 10 meters, it means the drone is 10 meters away from the target, either 10 meters east or 10 meters west. Absolute values help engineers define this 'tolerance' or acceptable error range.

Key Vocabulary

Key Terms

Absolute Value: The distance of a number from zero, always positive or zero. | Equation: A mathematical statement showing two expressions are equal. | Variable: A symbol (usually a letter) representing an unknown value. | Solution: The value(s) of the variable that make the equation true.

What's Next

What to Learn Next

Great job understanding absolute value equations! Next, you can explore solving absolute value inequalities, which use greater than (>) or less than (<) signs. This will help you define ranges of values, which is super useful in real-world problems like setting temperature limits or ingredient quantities.