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**Mathematics for Business and Economics - I**

Chapter 0. Algebra Refreshers

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**Sec0.3 Exponents and Radicals**

The product is abbreviated In general; for any positive integer n EXPONENT n - factors BASE

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**Basic Laws of exponents**

Rule 1 : Example 1: Example 2: Rule 2 : Example :

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**Basic Laws of exponents**

Rule 3 : Example : Rule 4 : Example 1: Example 2:

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**Basic Laws of exponents**

Rule 5 : Example 1: Example 2:

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**Basic Laws of exponents**

Rule 6 : Example : Rule 7 : Example 1: Example 2:

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**Basic Laws of exponents**

Rule 8 : Example : Rule 9 : Example 1: Example 2:

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Example : Example : Example : Example : Example : Example : Example : Example :

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0.5 FACTORING

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0.6 FRACTIONS

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If we add or subtract two fraction having a common denominator then the resulting is a fraction whose denominator is the common denominator

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LINEAR EQUATIONS A linear equation in one variable, such as x, is an equation that can be written in the standard form where a and b are real numbers with a ≠ 0.

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**FINDING THE SOLUTION: Procedure for solving linear equations in one variable**

Step 1 Eliminate Fractions. Multiply both sides of the equation by the least common denominator (LCD) of all the fractions. Step 2 Simplify. Simplify both sides of the equation by removing parentheses and other grouping symbols (if any) and combining like terms.

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**FINDING THE SOLUTION: Procedure for solving linear equations in one variable**

Step 3 Isolate the Variable Term. Add appropriate expressions to both sides, so that when both sides are simplified, the terms containing the variable are on one side and all constant terms are on the other side. Step 4 Combine Terms. Combine terms containing the variable to obtain one term that contains the variable as a factor.

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**FINDING THE SOLUTION: Procedure for solving linear equations in one variable**

Step 5 Isolate the Variable. Divide both sides by the coefficient of the variable to obtain the solution. Step 6 Check the Solution. Substitute the solution into the original equation.

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**Solve: Step 2 Step 3 Step 4 Solution EXAMPLE 1**

Solving a Linear Equation Solve: Solution Step 2 Step 3 Step 4

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**Solve: Step 5 Step 6 Check: The solution set is {3}.**

EXAMPLE 1 Solving a Linear Equation Solve: Solution continued Step 5 Step 6 Check: The solution set is {3}.

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EXAMPLE 2 Solving a Linear Equation Solve: Solution Step 2 Step 3

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EXAMPLE 2 Solving a Linear Equation Solve: Solution continued 0 = 0 is equivalent to the original equation. The equation 0 = 0 is always true and its solution set is the set of real numbers. So the solution set to the original equation is the set of real numbers. The original equation is an identity.

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EXAMPLE 3 Solving a Linear Equation Solve: x – 9x = 12x -5 Solution: –6x = 12x – x –12x = 12x –12x x = 2 The solution set is { }. EXAMPLE 4 Solving a Linear Equation Solve Solution: Since the LCD of 2 and 3 is 6, we multiply both sides of the equation by 6 to clear of fractions. Cancel the 6 with the 2 to obtain a factor of 3, and cancel the 6 with the 3 to obtain a factor of 2. Distribute the 3. Combine like terms. The solution set is { }.

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**Solve: 6 – (4 +x) = 8x – 2(3x + 5) Solution: Step 1: Does not apply**

EXAMPLE 5 Solving a Linear Equation Solve: – (4 +x) = 8x – 2(3x + 5) Solution: Step 1: Does not apply Step 2: – 4 –x = 8x – 6x –10 -x +2 = 2x – 10 Step 3: -x + (-2x) +2 +(-2) = 2x + (-2x) –10 + (-2) -3x = -12 Step 4: Step 5: 6 – (4 + 4) = 8(4) – 2(3(4) + 5) 6 – 8 = -2 = -2 The solution set is {4}.

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**Solve: Solution: Step 1: Step 2: Step 3: Step 4: Step 5:**

EXAMPLE 6 Solving a Linear Equation Solve: Solution: Step 1: Step 2: Step 3: Step 4: Step 5: The solution set is {-1}.

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EXAMPLE 7 Solving a Linear Equation

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EXAMPLE 8 Solving a Linear Equation

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EXAMPLE 8 Solving a Linear Equation

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**we raise both sides to power 2.**

EXAMPLE 4 Solving Equations Involving Radicals Solve: Solution Since we raise both sides to power 2.

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**P P Check each solution. ? ? ? –3 is an extraneous solution.**

EXAMPLE 4 Solving Equations Involving Radicals Solution continued Check each solution. ? P ? P ? –3 is an extraneous solution. The solution set is {0, 2}.

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**Isolate the radical on one side.**

EXAMPLE 5 Solving Equations Involving Radicals Solve: Solution Step 1 Isolate the radical on one side. Step 2 Square both sides and simplify.

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**P Step 3 Set each factor = 0. Step 4 Check. ? ?**

EXAMPLE 5 Solving Equations Involving Radicals Solution continued Step 3 Set each factor = 0. Step 4 Check. ? P ? 0 is an extraneous solution. The solution set is {4}.

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**Isolate one of the radicals.**

EXAMPLE 6 Solving an Equation Involving Two Radicals Solve: Solution Step 1 Isolate one of the radicals. Step 2 Square both sides and simplify.

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EXAMPLE 6 Solving an Equation Involving Two Radicals Solution continued Step 3 Repeat the process - isolate the radical, square both sides, simplify and factor.

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**P P Step 4 Set each factor = 0. Step 5 Check. ? ?**

EXAMPLE 6 Solving an Equation Involving Two Radicals Solution continued Step 4 Set each factor = 0. Step 5 Check. P ? P ? The solution set is {1,5}.

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QUADRATIC EQUATION A quadratic equation in the variable x is an equation equivalent to the equation where a, b, and c are real numbers and a ≠ 0. A quadratic equation is also called a second- degree equation.

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**THE ZERO-PRODUCT PROPERTY**

Let A and B be two algebraic expressions. Then AB = 0 if and only if A = 0 or B = 0.

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**SOLVING A QUADRATIC EQUATION BY FACTORING**

Step 1 Write the given equation in standard form so that one side is 0. Step 2 Factor the nonzero side of the equation from Step 1. Step 3 Set each factor obtained in Step 2 equal to 0. Step 4 Solve the resulting equations in Step 3. Step 5 Check the solutions obtained in Step 4 in the original equation.

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**The solutions check in the original equation.**

EXAMPLE 1 Solving a Quadratic Equation by Factoring. Solve by factoring: Solution The solutions check in the original equation. The solution set is

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**The solutions check in the original equation.**

EXAMPLE 2 Solving a Quadratic Equation by Factoring. Solve by factoring: Solution The solutions check in the original equation. The solution set is

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**Solve by factoring: EXAMPLE 3**

Solving a Quadratic Equation by Factoring. Solve by factoring:

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**Check the solution in original equation. Step 5**

EXAMPLE 3 Solving a Quadratic Equation by Factoring. Solve by factoring: Solution Step 1 Step 2 Step 3 Step 4 Check the solution in original equation. Step 5 The solution set is {4}. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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**THE SQUARE ROOT PROPERTY**

Suppose u is any algebraic expression and d ≥ 0.

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**Solve: The solution set is Solution**

Solving an Equation by the Square Root Method EXAMPLE 4 Solve: Solution The solution set is

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Solution Solution The equation has no roots

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