“ ��������������������������������������������������������������������������������������������������������fi�������������������������������������������������������������� ��������������������������������������������������������������������������������������������������������������fi����������” ������������������������������������������������������������������ �� ���������������� �������� Equations and Their Solutions In this chapter, we will learn how to solve equations and inequalities using two important properties—the addition property of equality and the multiplication property of equality. We will also learn real-life applications of these equations and inequalities. We will rely heavily on the skills we obtained from Chapter 1. From June 2009 to June 2010, Oprah Winfrey and James Cameron, director of the movies Avatar and Titantic, were the two highest paid celebrities. They earned a combined total of $525 million. Winfrey earned $105 million more than Cameron. How much did each of them earn? (Source: http://www.forbes.com/lists/) In this section, we will learn how to express this type of information as a mathematical equation. We will learn how to solve these types of equations in Section 2.2. Equations and Expressions In Chapter 1, we worked with numerical expressions and algebraic expressions. In this section, we turn our attention to equations. An equation is a mathematical statement that two expressions are equal. Equations can be numerical or algebraic. Some examples of each are shown in the following table. Numerical Equations Algebraic Equations 2 + 3 = 5 -7 + 6 = 1 x + 4 = 2 x2 = x Numerical equations are either true or false. In the above example, 2 + 3 equals 5, so this equation is true. The next example is false since -7 + 6 does not equal 1. Algebraic equations are neither true nor false until we assign a value to the variable. The value(s) of the variable that makes the equation true is called a solution of the equation. It is helpful to understand the difference between an equation and an expression. Equations are solved but expressions are simplified. ������������������������ Equations and Expressions 1. An expression consists of terms that are combinations of letters and numbers. Recall from Section 1.8 that terms are connected by addition or subtraction signs. 2. If two expressions are connected by an equals sign, then it is an equation. �� ������������������������ ������������������ Determine if each problem is an expression or equation. Problems Solutions 1a. 2x + 4 - 6x + 3 2x + 4 - 6x + 3 is an expression because it does not contain an equals sign. 1b. 2x + 4 = 6x + 3 2x + 4 = 6x + 3 is an equation because it contains an equals sign. 1c. 4y2 + 2y = 1 4y2 + 2y = 1 is an equation because it contains an equals sign. 1d. a2 + (a + 1)2 - 6 a2 + (a + 1)2 - 6 is an expression because it does not contain an equals sign. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Identify an equation and expression. 2. Determine if a number is a solution of an equation. 3. Express statements as mathematical equations. 4. Set up equations for application problems. 5. Troubleshoot common errors. ������������������������ ▶ Identify an equation and expression. 92
hendricks_beginning_algebra_1e_ch1_3
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