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Absolute Values
Solving Two-Step Equations Algebraically
Multiplying Monomials
Factoring Trinomials
Solving Quadratic Equations
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
Polynomials
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
Point
Inequalities
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
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
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
Logarithms
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
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
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 | = a

We can use what we know about absolute value to solve equations that contain absolute value.

For example, let’s solve the equation |x| = 5.

The solutions are those numbers that are 5 units from 0 on the number line. So, the equation |x| = 5 has two solutions: x = 5 and x = -5.

Now let’s solve the equation |x| = 0.

The solution contains those numbers that are 0 units from 0.

So, the equation |x| = 0 has only one solution: x = 0.

Finally, let’s consider the equation |x| = -5. The solution contains those numbers that are -5 units from 0. Since distance is never negative, there are no numbers that satisfy this equation.

Thus, the equation |x| = -5 has no solutions.

We can generalize these ideas to find the solutions of |z| = a.

 

Principle

Absolute Value Equations of the Form | z| = a

When solving an equation of the form |z| = a:

If a > 0, there are two solutions, z = a and z = -a.

If a = 0, there is one solution, z = 0.

If a < 0, there are no solutions.

Here, z represents an algebraic expression and a represents a real number. z

 

Example

Solve each equation:

a. | x | = 37

b. | y | = 0

c. | w | = .4

Solution

a. |x| = 37 has the form |z| = a where a = 37, a positive number.

So, the solutions of |x| = 37 are x = -37 and x = 37.

b. |y| = 0 has the form |z| = 0.

So, the solution of |y| = 0 is y = 0.

c. |w| = -4 has no solution because the absolute value of a number cannot be negative.

Let’s solve another equation involving absolute value: |9y| = 63
First, let’s rewrite the equation by substituting z for 9y. |z| = 63
This equation has two solutions. z = 63 and z = -63
Now, to solve for y, we substitute back 9y for z. 9y = 63 and 9y = -63
Divide both sides by 9. y = 7 and y = -7

So, the solutions of |9y| = 63 are y = 7 and y = -7.

We can solve equations containing one absolute value expression using the following procedure.

 

Procedure

To Solve an Equation Containing One Absolute Value Expression

Step 1 Isolate the absolute value expression to get an equation of the form |ax + b| = c.

Step 2 Make the substitution z = ax + b.

Step 3 Use the Absolute Value Principle to solve for z.

Step 4 Replace z with ax = b.

Step 5 Solve for x.

Step 6 Check the answer.

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