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Absolute Values
Solving Two-Step Equations Algebraically
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Factoring Trinomials
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Power Functions and Transformations
Composition of Functions
Rational Inequalities
Equations of Lines
Graphing Logarithmic Functions
Elimination Using Multiplication
Multiplying Large Numbers
Multiplying by 11
Graphing Absolute Value Inequalities
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Graphing Equations in Three Variables
Properties of Exponents
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Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
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Solving Absolute Value Equations

Solving an Equation of the Form | z| = |w|

The equation |z| = |w| says that the distance on a number line from z to 0 the same as the distance from w to 0. So, z and w must be equal or they must be opposites.

This can be summarized in the following principle:


Absolute Value Equations of the Form | z| = |w|

In the equation |z| = |w|,

let z and w represent any algebraic expressions.

Then, either z and w are equal: z = w or z and w are opposites: z = -w.



If z and w are equal, then they are at the same location on a number line.

If z and w are opposites, then they are the same distance from zero on a number line, but on opposite sides of 0.

We will use this principle to solve this equation:

Replace x + 6 with z and 2x with w.

Rewrite the absolute value equation as two equations.

Replace z with x + 6 and w with 2x.

Subtract x from both sides.

In the equation on the right, divide both sides by -3.

|x + 6| = |2x|

|z| = |w|

z = w or z = -w

x + 6 = 2x or x + 6 = -2x

6 = x or 6 = -3x

6 = x or -2 = x

Thus, the solutions to |x + 6| = |2x| are x = 6 and x = -2.
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