Free Algebra
Arithmetic Operations with Numerical Fractions
Multiplying a Polynomial by a Monomial
Solving Linear Equation
Solving Linear Equations
Solving Inequalities
Solving Compound Inequalities
Solving Systems of Equations Using Substitution
Simplifying Fractions 3
Factoring quadratics
Special Products
Writing Fractions as Percents
Using Patterns to Multiply Two Binomials
Adding and Subtracting Fractions
Solving Linear Inequalities
Adding Fractions
Solving Systems of Equations -
Exponential Functions
Integer Exponents
Example 6
Dividing Monomials
Multiplication can Increase or Decrease a Number
Graphing Horizontal Lines
Simplification of Expressions Containing only Monomials
Decimal Numbers
Negative Numbers
Subtracting Polynomials
Adding and Subtracting Fractions
Powers of i
Multiplying and Dividing Fractions
Simplifying Complex Fractions
Finding the Coordinates of a Point
Fractions and Decimals
Rational Expressions
Solving Equations by Factoring
Slope of a Line
Percent Introduced
Reducing Rational Expressions to Lowest Terms
The Hyperbola
Standard Form for the Equation of a Line
Multiplication by 75
Solving Quadratic Equations Using the Quadratic Formula
Raising a Product to a Power
Solving Equations with Log Terms on Each Side
Monomial Factors
Solving Inequalities with Fractions and Parentheses
Division Property of Square and Cube Roots
Multiplying Two Numbers Close to but less than 100
Solving Absolute Value Inequalities
Equations of Circles
Percents and Decimals
Integral Exponents
Linear Equations - Positive and Negative Slopes
Multiplying Radicals
Factoring Special Quadratic Polynomials
Simplifying Rational Expressions
Adding and Subtracting Unlike Fractions
Graphuing Linear Inequalities
Linear Functions
Solving Quadratic Equations by Using the Quadratic Formula
Adding and Subtracting Polynomials
Adding and Subtracting Functions
Basic Algebraic Operations and Simplification
Simplifying Complex Fractions
Axis of Symmetry and Vertices
Factoring Polynomials with Four Terms
Evaluation of Simple Formulas
Graphing Systems of Equations
Scientific Notation
Lines and Equations
Horizontal and Vertical Lines
Solving Equations by Factoring
Solving Systems of Linear Inequalities
Adding and Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Fractions
Solving Linear Equations
Simple Trinomials as Products of Binomials
Solving Nonlinear Equations by Factoring
Solving System of Equations
Exponential Functions
Computing the Area of Circles
The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
Squaring a Difference
Changing the Sign of an Exponent
Adding Fractions
Powers of Radical Expressions
Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
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Solving Equations


Example 6




We can multiply both sides by any nonzero number we like. Since is the reciprocal of , we multiply each side by .

Using the multiplication principle: Multiplying both sides by “eliminates” the on the left.

1x = 8 Simplifying

x = 8 Using the identity property of 1


The solution is 8.

In Example 6, to get x alone, we multiplied by the reciprocal, or multiplicative inverse of . We then simplified the left-hand side to x times the multiplicative identity, 1, or simply x. These steps effectively replaced the on the left with 1.

Because division is the same as multiplying by a reciprocal, the multiplication principle also tells us that we can “divide both sides by the same nonzero number”. That is,

if a = b then (provided c 0 ).

In a product like 3x , the multiplier 3 is called the coefficient. When the coefficient of the variable is an integer or a decimal, it is usually easiest to solve an equation by dividing on both sides. When the coefficient is in fraction notation, it is usually easier to multiply by the reciprocal.

Example 7




Using the multiplication principle: Dividing both sides by -4 is the same as multiplying by

1x = -23 Simplifying

x = -23 Using the identity property of 1


The solution is -23.


Dividing both sides by 3 or multiplying both sides by



The solution is 4.2.

c) To solve an equation like -x = 9 remember that when an expression is multiplied or divided by -1 its sign is changed. Here we multiply both sides by -1 to change the sign of -x :

-x = 9

(-1)(-x) = (-1)9 Multiplying both sides by -1 (Dividing by -1 would also work)

x = -9 Note that (-1)(-x) is the same as (-1)(-1)x


The solution is -9.

d) To solve an equation like we rewrite the left-hand side as and then use the multiplication principle:

Rewriting as

Multiplying both sides by

Removing a factor equal to

y = 12


The solution is 12.

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