166 Chapter 2 Linear Equations and Inequalities in One Variable �� ������������������������ Linear Inequalities in One Variable ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Graph the solution set of an inequality. 2. Write the solution set of an inequality in interval notation. 3. Write the solution set of an inequality in set-builder notation. 4. Solve linear inequalities. 5. Solve applications of linear inequalities. 6. Troubleshoot common errors. As math classes come to an end, a prevailing thought of many students is, “What grade do I need to make on the final to pass this class, or to keep my A?” In this section, we will look at one example of how to determine the grade needed on a final exam to achieve a certain grade in a class. This is an application of linear inequalities. Graphs of Inequalities Until now, we have solved only linear equations in one variable. We now turn our focus to solving linear inequalities in one variable. Linear equations and linear inequalities are similar except that a linear inequality has an inequality symbol instead of an equals sign. In Chapter 1, inequality symbols were introduced. �������������������������� A linear inequality in one variable is an inequality of the form ax + b < c, where a, b, and c are real numbers with a ≠ 0. Note that a linear inequality can have any of the inequality symbols: <, ≤, >, or ≥. Some examples of linear inequalities in one variable are 2x + 3 < -7 7x - 2 ≥ 6x + 4 4(3y - 1) < -2( y -9) Like linear equations, linear inequalities are neither true nor false until a value is assigned to the variable. Our job is to determine the values of the variable that make the inequality true. Each such value is a solution of a linear inequality. Linear inequalities generally have infinitely many values that make the statement true. Because we will not be able to list all of these values, we graphically represent the solution set of all the solutions on a number line. So, before we actually solve a linear inequality, we will examine the graphs of possible solution sets. ���������������������� Graphing the Solution Set of a Linear Inequality Step 1: Draw a number line. Step 2: Shade the portion of the number line that satisfies the inequality. Step 3: Put the appropriate symbol on the endpoint. a. If the endpoint satisfies the inequality, then it is included in the solution set as denoted by a bracket. A closed circle can also be used in place of the bracket. b. If the endpoint does not satisfy the inequality, then it is not included in the solution set as denoted by a parenthesis. An open circle can also be used in place of the parentheses. �� ������������������������ ������������������ Graph the solution set of each inequality on a number line. 1a. x > 3 1b. x ≤ 4 1c. 1 2 < x 1d. -2 < x ≤ 5 Solutions 1a. To graph the solutions of x > 3, we find the values of x that are larger than 3. We know the integers 4, 5, 6, and so on are larger than 3, but we must not forget that there are numbers between 3 and 4 that are also larger than 3 (e.g., 3.9, 3.54, 3.0002, etc.). The number 3 itself is not a solution of this inequality since 3 is not greater than itself. So, we shade the portion of the number line that is to the right of 3. To indicate that 3 is not included in the solution set, put a parenthesis on it. 0 1 2 3 4 5 6 7 8 9 10 Objective 1 ▶ Graph the solution set of an inequality.
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