Understanding Absolute Value - Algebra I | Luna Learn 24

Understanding Absolute Value

Absolute value is a fundamental concept in algebra that measures the distance of a number from zero on the number line, regardless of direction. In this lesson, we'll explore how to find and work with absolute values, including solving equations and inequalities involving absolute values.

Outline

Introduction to Absolute Value
Properties of Absolute Value
Solving Absolute Value Equations
Solving Absolute Value Inequalities
Examples and Practice Exercises
Homework
Revision
YouTube Video Explanation

Examples

Example 1

Find the absolute value of \( | -7 | \).

The absolute value of \( -7 \) is \( 7 \), because the distance of -7 from zero is 7 units.

Example 2

Solve the equation \( | x | = 5 \).

For \( | x | = 5 \), the solutions are \( x = 5 \) and \( x = -5 \), because both values are 5 units away from zero.

Example 3

Solve the equation \( | x - 3 | = 4 \).

To solve \( | x - 3 | = 4 \), we set up two equations: \( x - 3 = 4 \) and \( x - 3 = -4 \). Solving these gives \( x = 7 \) and \( x = -1 \).

Example 4

Solve the inequality \( | x | \leq 3 \).

For \( | x | \leq 3 \), the solution set is \( -3 \leq x \leq 3 \), as x can be any value within this range.

Example 5

Solve the inequality \( | 2x + 1 | > 5 \).

To solve \( | 2x + 1 | > 5 \), we set up two inequalities: \( 2x + 1 > 5 \) and \( 2x + 1 < -5 \). Solving these gives \( x > 2 \) and \( x < -3 \).

Example 6

Solve the equation \( | 3x - 4 | = 7 \).

To solve \( | 3x - 4 | = 7 \), we set up two equations: \( 3x - 4 = 7 \) and \( 3x - 4 = -7 \). Solving these gives \( x = 11/3 \) and \( x = -1 \).

Exercises

Exercise 1

Find the absolute value of \( | 8 | \).

The absolute value of \( 8 \) is \( 8 \).

Exercise 2

Solve the equation \( | x + 2 | = 6 \).

Solutions are \( x = 4 \) and \( x = -8 \).

Exercise 3

Solve the inequality \( | x - 1 | < 2 \).

The solution set is \( -1 < x < 3 \).

Exercise 4

Solve the inequality \( | 4x + 2 | \geq 8 \).

The solution set is \( x \leq -2.5 \) or \( x \geq 1.5 \).

Exercise 5

Solve the equation \( | 5 - 2x | = 3 \).

Solutions are \( x = 1 \) and \( x = 4 \).

Exercise 6

Solve the inequality \( | -3x + 7 | < 4 \).

The solution set is \( 1 < x < 3.67 \).

Homework

Complete the following exercises for additional practice:

Find the absolute value of \( | -12 | \)
Solve the equation \( | 2x + 5 | = 10 \)
Solve the inequality \( | x - 4 | \leq 3 \)
Solve the inequality \( | 3x + 1 | > 2 \)

Revision

Review the following key points:

Absolute value measures the distance of a number from zero, ignoring the direction.
To solve absolute value equations, create and solve two separate equations.
To solve absolute value inequalities, split into two cases based on the inequality sign.

Video Explanation

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