2. CHAPTER R: Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
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3. R.5 The Basics of Equation Solving
Solve linear equations.
Solve quadratic equations.
Solve a formula for a given letter.
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Equations
An equation is a statement that two expressions are equal.
To solve an equation in one variable is to find all the values of the variable that make the equation true.
Each of these numbers is a solution of the equation. The set of all solutions of an equation is its solution set. Equations that have the same solution sets are called equivalent equations.
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Equations
A linear equation in one variable is an equation that is equivalent to one of the form ax + b = 0, where a and b are real numbers and a  0.
A quadratic equation is an equation that is equivalent to one of the form ax2 + bx + c = 0, where a, b and c are real numbers and a  0.
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6. Equation- Solving Principles
For any real numbers a, b, and c,
The Addition Principle: If a = b is true, then a + c = b + c is true.
The Multiplication Principle: If a = b is true, then ac = bc is true.
The Principle of Zero Products: If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or b = 0, then ab = 0.
The Principle of Square Roots: If x2 = k, then
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Examples
Solve.
8x = 2(18 – 2x)
8x = 36 – 4x
12x = 36
x = 3
The solution checks.
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Examples
Solve.
x2 – 8x = – 16
x2 – 8x + 16 = 0
(x – 4 )(x – 4) = 0
x – 4 = 0 or x – 4 = 0
x = 4 or x = 4
There is only one solution, 4.
The solution checks.
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9. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Solve.
The solution checks.
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10. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Formulas
A formula is an equation that can be used to model a situation.
We have used the motion formula, d = r • t.
We have used the simple-interest formula, I = Prt.
We can use the equation-solving principles to solve a formula for a given variable.
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Example
Solve P = 2l + 2w for l.
Solution:
The formula
can be used to determine a
rectangle’s length if we know its perimeter and width.
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