ࡱ> ZXuC A U@ tbjbj Tk]KKKK@o@o@o8xo4tK-K:wXKl8IIIIIII$gMROrIKI%FJ0m m m Lb+\KIm Im rm  *2;K4w [@K@o; BdJ<-K+Pd +P4KK+PK4 m IIKK bG& KK  EMBED Equation.DSMT4  Chapter 0 Algebra Preliminaries Calculus is the reformulation of elementary mathematics through the use of a limit process. Students who have weak elementary mathematic skills may have a difficult time learning calculus. In this chapter, we first provide a pretest. If the result of your pretest is above 70% of the total score, your elementary mathematics may be sufficient. Then we will review some basic algebraic formulas and skills which are absolutely necessary for students to know before proceeding to calculus courses. After the review, we will provide a posttest to see how much you have improved your algebraic skills. Finally, we will study the results of the posttest. Pretest__The following problems will be demonstrated in the review. 1. Simplify and eliminate the radical by writing the given expression with fractional exponents:  EMBED Equation.DSMT4  2. Simplify:  EMBED Equation.DSMT4  3. Factor:  EMBED Equation.DSMT4  4. Multiply:  EMBED Equation.DSMT4  5. Simplify:  EMBED Equation.DSMT4  6. Solve for x:  EMBED Equation.DSMT4  7. Rationalize the numerator:  EMBED Equation.DSMT4  EMBED Equation.DSMT4  8. For  EMBED Equation.DSMT4 , find and simplify:  EMBED Equation.DSMT4  9. Find all values of x that satisfy  EMBED Equation.DSMT4 . 10. Rewrite EMBED Equation.DSMT4  as a piecewise-defined function. Algebra Review 0.1 Properties of Exponents An expression of the form  EMBED Equation.DSMT4  is called an exponential expression, where the variable a is called the base, and n is called the exponent or power to which the base is raised. Any exponential expression with a rational exponent can be written as a radical expression. The following table provides the properties of exponents that are frequently used in simplifying certain expressions. ( Table 0.1 Properties of Exponents  EMBED Equation.DSMT4  Let a and b be positive numbers and let m and n be real numbers. Then,1.  EMBED Equation.DSMT4 2.  EMBED Equation.DSMT4 3.  EMBED Equation.DSMT4 4.  EMBED Equation.DSMT4 5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  ( Example 0.1 Use the properties of exponents to simplify the following expressions. Write the answers so that radical notions are not used and there are no variables in the denominator. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  ( Solution (a)   EMBED Equation.DSMT4  (b) We need to eliminate the radicals.  EMBED Equation.DSMT4  Note:  EMBED Equation.DSMT4  (c) We only need to eliminate the variables from the denominator.  EMBED Equation.DSMT4  (d) We need to eliminate the radical notion and variables from the denominator.  EMBED Equation.DSMT4  ( Rationalizing the Denominator There are times when it is convenient to remove a radical from the denominator of fractions. For example, we want to rewrite the fraction  EMBED Equation.DSMT4 in an equivalent form, but without the radical in the denominator. We will multiply  EMBED Equation.DSMT4  by  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4  The procedure used to remove a radical from the denominator is called rationalizing the denominator. To simplify a fraction which has radicals in the denominator, multiply the numerator and denominator by the conjugate of the denominator. For example, the expressions  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are conjugates of each other. ( Example 0.2 (a) Rationalize the denominator and simplify:  EMBED Equation.DSMT4  (b) Rationalize the numerator and simplify:  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ) ( Solution (a) After multiplying the numerator and denominator by the conjugate of  EMBED Equation.DSMT4 , we need use the formula  EMBED Equation.DSMT4  to simplify the denominator.  EMBED Equation.DSMT4  (b) In calculus, it is sometimes necessary to rationalize a numerator. The same techniques as in (a) can be used to rationalize the numerator of a fraction.   EMBED Equation.DSMT4  ( 0.2 Operations on Fractions Recall that a fraction has a form of  EMBED Equation.DSMT4  , where a and b can be integers or algebraic expressions. If a and b (b `" 0) are integers,  EMBED Equation.DSMT4  is called a rational number. For example,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are rational numbers. If a and b are algebraic expressions,  EMBED Equation.DSMT4  is called a fractional expression. For example,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are fractional expressions. The following table provides the operation rules for combining two (or more) fractions into one. ( Table 0.2 Operations on Fractions (The denominators in each case are assumed to be nonzero.) 1.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4 ( Example 0.3 Carry out the indicated operations and simplify: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) To add or subtract two fractions with different denominators, we need to find the least common denominator (LCD). Since  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , the LCD is  EMBED Equation.DSMT4 = EMBED Equation.DSMT4 . Multiply the first term by  EMBED Equation.DSMT4 and the second term by  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4  EMBED Equation.DSMT4  (b)   EMBED Equation.DSMT4  (c) This type of expression occurs in calculus.   EMBED Equation.DSMT4  An alternate way to solve the problem is by multiplying the given expression by  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is the LCD for the fractions in the numerator.  EMBED Equation.DSMT4  Which way is better? ( 0.3 Operations on Polynomials Adding and Subtracting Polynomials An expression of the form  EMBED Equation.DSMT4  is called a Polynomial, where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 are real numbers and  EMBED Equation.DSMT4 . The degree of a polynomial is the highest power of the variable. Thus,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 are polynomials of degree 1, 2 and 4, respectively. Polynomials can be added by combining similar (or like) terms. For example,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are similar (because they have the same power of x), hence,  EMBED Equation.DSMT4 . But,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are not similar (because they have different powers of x), therefore, they cannot be combined. To subtract one polynomial from another, add the opposite of each term of the polynomial you are subtracting. ( Example 0.4 Take the indicated operations for the following: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solution  (a) (b)   EMBED Equation.DSMT4  ( Multiplication of Polynomials The product of two polynomials is the polynomial obtained by multiplying each term of one polynomial by each term of the other polynomial and then combining like terms. ( Example 0.5 Multiply: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solution (a) Multiplication of polynomials can be carried out by the distributive properties.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (b) A binomial is a polynomial that has two terms. Here,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are two binomials. It is frequently necessary to find the product of two binomials. The product can be found by using a method called FOIL, which is based upon the Distributive Property. The letters of FOIL stand for First, Outer, Inner, and Last.  Factoring Polynomials A polynomial of degree greater than 0 is said to be prime relative to a given set of numbers if all of its coefficients are from that set of numbers and it can not be written as a product of two polynomials, excluding 1 and itself, have coefficients from that set of numbers. For example, to the set of integer,  EMBED Equation.DSMT4 is a prime polynomial, but  EMBED Equation.DSMT4 is not prime, because  EMBED Equation.DSMT4 . A polynomial is said to be factored completely relative to a given set of number if it can be written as a product of prime polynomials relative to that set of numbers. The set of numbers most frequently used to factoring polynomials is the set of integers. Here, we will review some of the basic factoring techniques for polynomials with integer coefficients. ( Table 0.3 Basic Factoring Techniques TechniqueExampleCommon Factors  EMBED Equation.DSMT4 Grouping EMBED Equation.DSMT4 Difference of Squares:  EMBED Equation.DSMT4  EMBED Equation.DSMT4 Difference of Cubes:  EMBED Equation.DSMT4  Sum of Cubes:  EMBED Equation.DSMT4  EMBED Equation.DSMT4 Trial and Error for the form  EMBED Equation.DSMT4   EMBED Equation.DSMT4  This will be explained in example 0.6(c). ( Example 0.6 Factor each expression completely: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  (e)  EMBED Equation.DSMT4  ( Solution  (a) (b)  EMBED Equation.DSMT4  (c) Polynomials of the form  EMBED Equation.DSMT4 , where a, b, and c are real numbers with  EMBED Equation.DSMT4 , are called quadratic. The term  EMBED Equation.DSMT4 is the leading term,  EMBED Equation.DSMT4 is the linear term, and c is the constant term. The Trial and Error method is often used to factor a quadratic. For example, if  EMBED Equation.DSMT4 can be factored, we look for factors of the form  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , such that  EMBED Equation.DSMT4  Setting corresponding coefficients equal gives  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  These equations suggest a way to find k, s, u, and v. Since the coefficient of  EMBED Equation.DSMT4 is 1, EMBED Equation.DSMT4 . Therefore, both k and u equal 1. Since  EMBED Equation.DSMT4 , there are only four pair of factors possible:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 . Now we need to solve  EMBED Equation.DSMT4  to find out s and v. Since k and u are 1,  EMBED Equation.DSMT4 . Therefore choose the pair of factors that have a sum of -8: -3 and -5. This is the final answer:  EMBED Equation.DSMT4  We also can put  EMBED Equation.DSMT4  as a cross product form:  Then The result of the cross product is not the coefficient of the linear term. Try another pair.  The result of this cross product is the coefficient of the linear term. Therefore,  EMBED Equation.DSMT4 . You can use the same way to check that  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  dont give the right answer. (d) For  EMBED Equation.DSMT4 , the coefficient of  EMBED Equation.DSMT4 is 6. The possible factors of 6 are:  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The possible factors of the constant term -10 are:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . By doing all of possible cross products, only the following cross product gives the right coefficient of the linear term,  Therefore, the answer is  EMBED Equation.DSMT4 . (e) For  EMBED Equation.DSMT4 , since 1 is the coefficient of  EMBED Equation.DSMT4 , we need factors of -3 whose sum is 4. The only possible factors of -3 are (-1)(3) and (1)(-3). Neither pair of factors has sum 4. Therefore  EMBED Equation.DSMT4 cannot be factored over the set of integers and it is called irreducible over the integers. ( The method of trial and error illustrated in the above can be long and tedious, particularly if the coefficients of the polynomial are large and have many prime factors. But for simple cases, it is often possible to arrive at the correct choice very quickly. 0.4 Rational Expressions As we mentioned in section 0.1, a fractional expression is a quotient of two algebraic expressions. As a special case, if p and q are polynomials, the algebraic expression  EMBED Equation.DSMT4 is called a rational expression. The operations on fractions we reviewed in section 0.2 are also used for rational expressions. In algebra, as in arithmetic, we say that a fraction is reduced or simplified to lowest terms when the numerator and denominator contain no common factors (other than 1 and -1). The factoring techniques we reviewed in the previous section are used to simplify rational expressions. ( Example 0.7 Reduce the following expressions to lowest terms: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a)   EMBED Equation.DSMT4   (b)  (c) ( ( Example 0.8 For what values is the following rational expression undefined?  EMBED Equation.DSMT4  ( Solution Rational expressions name a real number for each real number replacement of the variable or variables. Thus all properties of real numbers apply to rational expressions. Since a real number divided by zero is undefined, the values that make a rational expression undefined are the values that make the denominator of the expression zero. In this case,  EMBED Equation.DSMT4  Thus when x equals -1 or 2, the rational expression is undefined. ( 0.5 Difference Quotient Recall that a function is a rule that takes certain number as inputs and assigns to each input one and only one number as output. For example, the function  EMBED Equation.DSMT4  takes each real number as inputs and square each input as an output. It is helpful to write the function  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4 , which means whatever quantity goes in the parentheses on the left hand side, must also be placed in the parentheses on the right. ( Example 0.9 Given that  EMBED Equation.DSMT4 , find and simplify the following expressions: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) The number 5 is the input for the given function, we need find the output  EMBED Equation.DSMT4 .   EMBED Equation.DSMT4  (b) The 2b is the input, we need find the output  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (c) The  EMBED Equation.DSMT4 is the input, we need find the out put  EMBED Equation.DSMT4 .   EMBED Equation.DSMT4  Note:  EMBED Equation.DSMT4  and also  EMBED Equation.DSMT4  In this case,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  ( A special fractional expression of a function, called the difference quotient, plays an important role in calculus. For now, we will concentrate on the algebraic techniques used in calculating and simplifying such expressions. You will learn the meaning and applications later when you learn calculus. ( Table 0.4 Definition of The Difference Quotient If f is any function and h is any nonzero number, then the difference quotient of f is the expression  EMBED Equation.DSMT4  ( Example 0.10 Given that  EMBED Equation.DSMT4 , find and simplify  EMBED Equation.DSMT4 . ( Solution From the example 0.9 (c), we have already found EMBED Equation.DSMT4 , so   EMBED Equation.DSMT4  ( ( Example 0.11 Compute the difference quotient of  EMBED Equation.DSMT4  when x = 1. Rationalize the numerator if needed. ( Solution By the definition, the difference quotient when x = 1 is the expression  EMBED Equation.DSMT4 . Recall that  EMBED Equation.DSMT4  Therefore   EMBED Equation.DSMT4  ( ( Example 0.12 Compute the difference quotient of  EMBED Equation.DSMT4  when t = 2. ( Solution By the definition, the difference quotient when t = 2 is the expression  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Therefore,  EMBED Equation.DSMT4  The right hand side is the expression in Example 0.3 (c) that we have already simplified. ( 0.6 Solving Linear Equations A linear equation in one variable is an equation that can be written in the form  EMBED Equation.DSMT4 , where a and b are real numbers and  EMBED Equation.DSMT4 . A solution (root) set of an equation is the set of real numbers which make the equation true. To solve an equation is to find all its solutions. Two equations are equivalent if they have the same solution set. To solve equations, we use the idea that adding or subtracting the same quantity on both sides of the equation and multiplying or dividing both sides of an equation by the same nonzero quantity produces an equivalent equation to generate a sequence of equivalent equations until we obtain one whose solution is obvious. ( Example 0.13 Solving linear equations: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution  (a) Thus the solution is 4. You can check this answer by substituting 4 for x in  EMBED Equation.DSMT4 . Check:  EMBED Equation.DSMT4  Thus, the answer is right. (b)  Thus the solution is 15. You can check this answer by substitute 15 for x in  EMBED Equation.DSMT4 . Check:  EMBED Equation.DSMT4  The answer is right. (c) When we multiply both sides of an equation by an expression containing a variable, we sometimes produce an equation that is not equivalent to the original equation. In this case, to eliminate the denominator, we need multiply both sides by  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4  Since the denominator is 0 when  EMBED Equation.DSMT4 , the number -3 does not satisfy the original equation. Therefore, we say the equation has no solution. It is necessary and important to check the answers, particularly when you multiply an equation by an expression containing a variable. An alternate way to solve above equation is by cross-multiplication. The idea is if  EMBED Equation.DSMT4 , then multiply both sides by bd, give  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Which way is better? ( 0.7 Solving Quadratic Equations Solving Quadratic Equations by Factoring A quadratic equation is an equation that can be written in the form  EMBED Equation.DSMT4 , where a, b, and c are real numbers and  EMBED Equation.DSMT4 . One of the simplest techniques for solving quadratic equation involves factoring and applying the following theorem. ( Table 0.4 The Zero-Factor Theorem If A and B are real numbers, then EMBED Equation.DSMT4  if and only if  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  (or both). ( Example 0.14 Solve each equation by factoring. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solution (a)  The solutions are  EMBED Equation.DSMT4  and -2. You can check the answers by substituting  EMBED Equation.DSMT4  and -2 for x in the equation respectively.  (b) The answers are 0, -2 and 4, which means the equation has three solutions. But if you divide both sides by x at the beginning, you would have found only two solutions: -2 and 4. ( Solving Quadratic Equations by Completing the Square The previous method requires the quadratic equation to be factored into integers. However, not every quadratic equation, like EMBED Equation.DSMT4 , can be factored into integers. A technique called completing the square can solve the problem. The idea of completing the square is using the algebraic skills to rewrite the form  EMBED Equation.DSMT4  into a form EMBED Equation.DSMT4 , which can be easily solved by taking the square root of each side. ( Example 0.15 Solve each equation by completing the square. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solution (a) Therefore the equation has two solutions:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . (b)  There are two solutions:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , which are called complex numbers. A complex number is a number of the form  EMBED Equation.DSMT4 , where a and b are real numbers. a is called the real part of the complex number and b is called the imaginary part of the complex number. To make it possible to solve all quadratic equations, mathematicians invented this complex number system. Example 0.15(b) has no solutions in the real number system, but have two solutions in the complex number system. Solving Quadratic Equations by Using the Quadratic Formula A general formula known as the quadratic formula can be derived by applying the method of completing the square to the standard form of a quadratic equation. This formula can be used to solve any quadratic equation. ( Table 0.4 The Quadratic Formula The solution of the general quadratic equation  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 , is  EMBED Equation.DSMT4  ( Example 0.16 Solve each equation by using the quadratic formula.. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) The equation EMBED Equation.DSMT4 is in standard form  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 , we have  EMBED Equation.DSMT4  The solution is a real number  EMBED Equation.DSMT4 and it is called a double root. (b) The equation EMBED Equation.DSMT4 is not in the standard form  EMBED Equation.DSMT4 , we need to rewrite it in standard form before using the formula.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The solutions are two real numbers:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . (c) The equation  EMBED Equation.DSMT4  is not in the standard form  EMBED Equation.DSMT4 , we need to rewrite it in standard form before using the formula.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The solutions are two complex numbers:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . The expression  EMBED Equation.DSMT4  in the quadratic formula is called the discriminant. Its value determines the nature of the roots of the given quadratic equation, as summarized in the following table. ( Table 0.5 The Discriminant  EMBED Equation.DSMT4  For any quadratic equation  EMBED Equation.DSMT4 , where a, b, and c are real numbers and  EMBED Equation.DSMT4 , If1.  EMBED Equation.DSMT4 The equation has two real and unequal roots.2.  EMBED Equation.DSMT4 The equation has one real root (double root).3.  EMBED Equation.DSMT4 The equation has two complex number (no real) roots.  0.8 Inequalities An inequality is a statement where two expressions are not equal. For example, 7 > 4 is an inequality which means 7 is greater than 4. The symbols > (greater than), < (less than), e" (greater than or equal), and d" (less than or equal), are used to write inequalities. A linear inequality in one variable is an inequality that can be written in the form  EMBED Equation.DSMT4 , where a, b, and c are real numbers, and  EMBED Equation.DSMT4 . The solution set of an inequality is the set of real numbers which make the inequality true. Most of properties we used for solving equations are also true for solving inequalities except for those involving multiplying and dividing each side of an inequality by a negative number. The basic properties are summarized in the following table. ( Table 0.6 Properties of Inequalities Let a , b and c be real numbers. 1. If  EMBED Equation.DSMT4  , then  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Example: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4 . 2. If  EMBED Equation.DSMT4  and c is positive, then  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4  and c is negative, then  EMBED Equation.DSMT4 . Example: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4 .3. If a < b and b < c , then  EMBED Equation.DSMT4 . Example: If  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .Above properties hold for other inequality relations: >,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 .  ( Example 0.15 Solve each inequality: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a)  The solution is a number set which includes all real numbers less than -3. The solution set can be denoted as x < -3 or the unbounded interval (-", -3). (b)  The solution is a number set which includes all real numbers greater than or equal to -4. The solution set can be denoted as x e" -4 or the unbounded interval [-4, "). (c) The solution is a number set which includes all real numbers greater than 4 and less than or equals to 5. The solution set can be denoted as  EMBED Equation.DSMT4  or the interval (4, 5]. 0.9 Absolute Value Recall that the absolute value of a number x, denoted by |x|, is the distance from x to 0 on the real number line. Distance is always positive or zero, so we have |x| e" 0 for every number x. ( Table 0.6 Definition of Absolute Value If x is a real number, then the absolute value of x is  EMBED Equation.DSMT4  Solving Absolute Value Equations and Inequalities To solve absolute value equations and inequalities, we use the following properties. ( Table 0.7 Properties of Absolute Value Equations and Inequalities If a > 0, then 1.  EMBED Equation.DSMT4 , is equivalent to   EMBED Equation.DSMT4 2.  EMBED Equation.DSMT4 , is equivalent to EMBED Equation.DSMT4 3.  EMBED Equation.DSMT4 , is equivalent to  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 The properties above also hold for the other inequality relations:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  ( Example 0.16 Solve: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  ( Solution (a) The equation  EMBED Equation.DSMT4  is equivalent to two equations:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , which can be solved separately:  Thus, the solutions are 6 and -1. (b) The inequality  EMBED Equation.DSMT4  is equivalent to the inequalities:  EMBED Equation.DSMT4 . Therefore,  EMBED Equation.DSMT4  The solution is the set  EMBED Equation.DSMT4 , or the interval  EMBED Equation.DSMT4 . (c) The inequality  EMBED Equation.DSMT4  is equivalent to the inequalities:  EMBED Equation.DSMT4 , which can be solved separately:  The solution are the sets x > 0 and x < -1, or the intervals (-", -1) and (0, "). ( Absolute Value Functions The definition of absolute value shown in Table 0.6 also defines a function which has a rule assigning each real number x its absolute value |x|. The function is called an absolute value function and the graph is shown in the following figure. ( Figure 0.1 Graph of the absolute value function: EMBED Equation.DSMT4   Note that the graph of  EMBED Equation.DSMT4  has a corner at x = 0 (origin) and has two straight lines:  EMBED Equation.DSMT4  for x e" 0 and  EMBED Equation.DSMT4  for x < 0. This kind of function which is defined for specific intervals is called a piecewise-defined function. ( Example 0.17 For the following functions, graph the function and rewrite the function in piecewise form. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  ( Solution (a) Since  EMBED Equation.DSMT4  has a corner when x = 0, to find the corner of  EMBED Equation.DSMT4 , we let  EMBED Equation.DSMT4 . The solution of  EMBED Equation.DSMT4 is -3, which means the corner of  EMBED Equation.DSMT4  has shifted 3 units to the left of the origin. See the figure below. Also if  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and if  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , thus   (b) To find the corner, we let  EMBED Equation.DSMT4 , then the solution  EMBED Equation.DSMT4  means the corner has shifted  EMBED Equation.DSMT4 units to the right of the right. If  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , therefore  (  Posttest 1. Simplify the following expression. The answer should not have radical or negative exponents.  EMBED Equation.DSMT4  2. For  EMBED Equation.DSMT4 , find the difference quotient when x = 2. 3. Solve by completing the square:  EMBED Equation.DSMT4  4. Factor:  EMBED Equation.DSMT4  5. Solve for x:  EMBED Equation.DSMT4  6. Solve for x:  EMBED Equation.DSMT4  7. Rationalize the numerator:  EMBED Equation.DSMT4  EMBED Equation.DSMT4  8. Simplify:  EMBED Equation.DSMT4  9. Find all values of x that satisfy  EMBED Equation.DSMT4 . 10. Rewrite  EMBED Equation.DSMT4  as a piecewise-defined function.  Analyzing Posttest Result The knowledge and skills in Algebra are essential for learning Calculus. The Pretest, Chapter 0, and the Posttest are all designed to refresh and review your Algebraic skills. However, they are not supposed to teach these skills to you. Posttest score (after review)Analysis90% or aboveYou are most likely prepared with the essential algebraic skills.Between 75% and 90%You are some what prepared with the essential Algebraic skills. It would be in your best interest to study on your own to improve your Algebra in order to avoid struggling when learning Calculus.75% or belowYou need to seriously reconsider taking this course. Your Algebraic skills do not seem to be adequate to take Calculus. Major difficulty exists. Please consult your advisor.  The numbers given are only guidelines. Please take all factors into account to make a mature and responsible decision on whether you should take this course or not. Answers to Pretest 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. x = 0, x = -2, x = 4 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9. [-1, 4] 10. EMBED Equation.DSMT4  Answers to Posttest 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. 10 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  or [3, ") 10. EMBED Equation.DSMT4  Set up two equations. Solve for x. Given. Factor.  EMBED Equation.DSMT4  if and only if  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . Solve for x. Rationalize the numerator. Simplify. Plug in EMBED Equation.DSMT4 in both sides. Apply  EMBED Equation.DSMT4  Distributive property Plug in 2b in both sides. Simplify. Plug in 5 in both sides..  EMBED Equation.DSMT4  Factor out the common factors from numerator and denominator Simplify. Rewrite as the form:  EMBED Equation.DSMT4  Apply:  EMBED Equation.DSMT4  Apply the Distributive Property  EMBED Equation.DSMT4 is a prime polynomial.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  or or  EMBED Equation.DSMT4   EMBED Equation.DSMT4  This line suggests EMBED Equation.DSMT4 . This line suggests EMBED Equation.DSMT4 . The first line suggests  EMBED Equation.DSMT4 . The second line suggests  EMBED Equation.DSMT4 . Given Distribute -2 to the second pair of parentheses. Group the similar terms together Simplify  EMBED Equation.DSMT4  Given  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are similar,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are similar. Group the similar terms together. Simplify.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Given Add 7 to each of the three parts of the inequality. Simplify. Multiply each of the three parts of the inequality by  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Given Subtract 7x from both sides. Subtract 1 from both sides. Multiply both sides by ( EMBED Equation.DSMT4 ) and reverse the Inequality symbol. Given Subtract both sides by 1. Simplify. Multiply both sides by ( EMBED Equation.DSMT4 ) and reverse the inequality symbol.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Given Subtract each side by 5. Divide each side by 9, making the coefficient of  EMBED Equation.DSMT4  equal to 1. Half of  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4 . Add the square of  EMBED Equation.DSMT4 to each side. Using  EMBED Equation.DSMT4 , the left hand side is  EMBED Equation.DSMT4 . Take the square root of each side. Simplify the right hand side. Note that  EMBED Equation.DSMT4 is not a real number, it is called an imaginary number and denoted as i ( EMBED Equation.DSMT4 ).  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Given Add 2 to each side. Half of the coefficient of x is 1 and  EMBED Equation.DSMT4 . Add 1 to each side. Using  EMBED Equation.DSMT4 , the left side is  EMBED Equation.DSMT4 . Take the square root of each side.  EMBED Equation.DSMT4 means  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . Set up two equations and solve for x.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Do not divide both sides by x. Subtract  EMBED Equation.DSMT4 from each side. To use the Zero-Factor theorem to solve the equation, one s(14IJ+ . 2 M [ \ r s   $ % ( : ? 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EMBED Equation.DSMT4  Factor out the common factors 2 and 3. The LCD is  EMBED Equation.DSMT4 . Simplify. By properties:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  Apply distributive property for numerator. Simplify the numerator. Factor out the common factor 13 from numerator Cancel  EMBED Equation.DSMT4 from numerator and denominator. 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