Learn on PengiCalifornia Reveal Math, Algebra 1Unit 1: Using Expressions and Equations

1-4 Absolute Value Expressions and Equations

In this Grade 9 lesson from California Reveal Math, Algebra 1, students learn to evaluate absolute value expressions and solve absolute value equations by considering two cases for nonnegative and negative values inside the absolute value symbols. The lesson uses real-world contexts such as thermometer accuracy and margin of error to connect absolute value to positive differences and distance on a number line. Students also explore special cases, including equations with no solution (the empty set), and graph solution sets on a number line.

Section 1

Writing Absolute Value Equations from Context

Property

When a quantity $x$ must be within a distance $d$ of a center value $m$ , the relationship is written as:

|x - m| = d

Section 2

Solving Absolute Value Equations: Two-Case Method

Property

The equation $|ax + b| = c$ (where $c > 0$ ) is equivalent to:

ax + b = c \quad \text{or} \quad ax + b = -c

Examples

To solve $|x - 3| = 8$ , we set up two equations: $x - 3 = 8$ or $x - 3 = -8$ . The solutions are $x = 11$ and $x = -5$ .
To solve $|2y + 5| = 11$ , we set up two equations: $2y + 5 = 11$ or $2y + 5 = -11$ . The solutions are $y = 3$ and $y = -8$ .
To solve $|\frac{z}{3} - 1| = 4$ , we set up two equations: $\frac{z}{3} - 1 = 4$ or $\frac{z}{3} - 1 = -4$ . The solutions are $z = 15$ and $z = -9$ .

Explanation

To solve an absolute value equation, you split it into two separate linear equations. This is because the expression inside the absolute value bars could be either positive or negative, and both would result in the same positive value.

Section 3

Absolute Value Equations: Isolating the Absolute Value First

Property

Before applying the two-case method to solve an absolute value equation, isolate the absolute value expression on one side of the equation. Once isolated, if $|A| = c$ where $c \geq 0$ , then:

A = c \quad \text{or} \quad A = -c

Book overview

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Unit 1: Using Expressions and Equations

Back to California Reveal Math, Algebra 1

Lesson 1
1-1 Properties of Real Numbers
Learn Practice
Lesson 2
1-2 Descriptive Modeling with Expressions
Learn Practice
Lesson 3
1-3 Solving Equations in One Variable
Learn Practice
Lesson 4Current
1-4 Absolute Value Expressions and Equations
Learn Practice
Lesson 5
1-5 Using Formulas
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Writing Absolute Value Equations from Context

Property

When a quantity $x$ must be within a distance $d$ of a center value $m$ , the relationship is written as:

|x - m| = d

Section 2

Solving Absolute Value Equations: Two-Case Method

Property

The equation $|ax + b| = c$ (where $c > 0$ ) is equivalent to:

ax + b = c \quad \text{or} \quad ax + b = -c

Examples

To solve $|x - 3| = 8$ , we set up two equations: $x - 3 = 8$ or $x - 3 = -8$ . The solutions are $x = 11$ and $x = -5$ .
To solve $|2y + 5| = 11$ , we set up two equations: $2y + 5 = 11$ or $2y + 5 = -11$ . The solutions are $y = 3$ and $y = -8$ .
To solve $|\frac{z}{3} - 1| = 4$ , we set up two equations: $\frac{z}{3} - 1 = 4$ or $\frac{z}{3} - 1 = -4$ . The solutions are $z = 15$ and $z = -9$ .

Explanation

Section 3

Absolute Value Equations: Isolating the Absolute Value First

Property

Before applying the two-case method to solve an absolute value equation, isolate the absolute value expression on one side of the equation. Once isolated, if $|A| = c$ where $c \geq 0$ , then:

A = c \quad \text{or} \quad A = -c

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 1: Using Expressions and Equations

Back to California Reveal Math, Algebra 1

Lesson 1
1-1 Properties of Real Numbers
Learn Practice
Lesson 2
1-2 Descriptive Modeling with Expressions
Learn Practice
Lesson 3
1-3 Solving Equations in One Variable
Learn Practice
Lesson 4Current
1-4 Absolute Value Expressions and Equations
Learn Practice
Lesson 5
1-5 Using Formulas
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Writing Absolute Value Equations from Context

Property

When a quantity $x$ must be within a distance $d$ of a center value $m$ , the relationship is written as:

|x - m| = d

Section 2

Solving Absolute Value Equations: Two-Case Method

Property

The equation $|ax + b| = c$ (where $c > 0$ ) is equivalent to:

ax + b = c \quad \text{or} \quad ax + b = -c

Examples

To solve $|x - 3| = 8$ , we set up two equations: $x - 3 = 8$ or $x - 3 = -8$ . The solutions are $x = 11$ and $x = -5$ .
To solve $|2y + 5| = 11$ , we set up two equations: $2y + 5 = 11$ or $2y + 5 = -11$ . The solutions are $y = 3$ and $y = -8$ .
To solve $|\frac{z}{3} - 1| = 4$ , we set up two equations: $\frac{z}{3} - 1 = 4$ or $\frac{z}{3} - 1 = -4$ . The solutions are $z = 15$ and $z = -9$ .

Explanation

Section 3

Absolute Value Equations: Isolating the Absolute Value First

Property

Before applying the two-case method to solve an absolute value equation, isolate the absolute value expression on one side of the equation. Once isolated, if $|A| = c$ where $c \geq 0$ , then:

A = c \quad \text{or} \quad A = -c

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 1: Using Expressions and Equations

Back to California Reveal Math, Algebra 1

Lesson 1
1-1 Properties of Real Numbers
Learn Practice
Lesson 2
1-2 Descriptive Modeling with Expressions
Learn Practice
Lesson 3
1-3 Solving Equations in One Variable
Learn Practice
Lesson 4Current
1-4 Absolute Value Expressions and Equations
Learn Practice
Lesson 5
1-5 Using Formulas
Learn Practice

1-4 Absolute Value Expressions and Equations

Property

Property

Examples

Explanation

Property

Unit 1: Using Expressions and Equations

1-1 Properties of Real Numbers

1-2 Descriptive Modeling with Expressions

1-3 Solving Equations in One Variable

1-4 Absolute Value Expressions and Equations

1-5 Using Formulas

Lesson overview

Property

Property

Examples

Explanation

Property

Unit 1: Using Expressions and Equations

1-1 Properties of Real Numbers

1-2 Descriptive Modeling with Expressions

1-3 Solving Equations in One Variable

1-4 Absolute Value Expressions and Equations

1-5 Using Formulas

1-1 Properties of Real Numbers

1-2 Descriptive Modeling with Expressions

1-3 Solving Equations in One Variable

1-4 Absolute Value Expressions and Equations

1-5 Using Formulas