Chapter 3 Graphing Fundamentals Section 3.1 Solving and Graphing a Linear Inequality of a Single Variable TERMINOLOGY 3.1 Previously Used: Isolate a Variable Simplifying Expressions Prerequisite Terms: Number Line Solution Set Solving Equations >, <,, New Terms to Learn: Closed Interval Formal definition Example 106
Multiplicative Law of Inequalities Formal definition Example Open Interval Formal definition Example READING ASSIGNMENT 3.1 Section 2. READING AND SELF-DISCOVERY QUESTIONS 3.1 1. What is a linear inequality? 2. What is the difference between solving a linear equation and solving a linear inequality? 3. How is the solution set of a linear inequality presented? 4. How do you validate a linear inequality? 107
Chapter 3 Graphing Fundamentals METHODOLOGY 3.1 SOLVING A LINEAR INEQUALITY The methodology highlights, in its steps, when and how to eliminate fractions, when and how to distribute out the parentheses, when to simplify the expressions, address negative coefficients, and denote symbolically and graphically the end points. The solution to such inequalities is usually an infinite number of values. Limitation/Caution: The need to reverse the direction of the inequality when multiplying or dividing by a negative number can be confusing. Solve for x: 1 ( x 3) < 1 (4x + 2) Solve for x: 2 2(4 x) > ( x + 3) 2 Steps Discussion 1 Clear the parentheses Use the Distributive Property in the clearing the parentheses technique 1 1 1 1 x 3 < 4 x + 2 2 3 2 Eliminate Fractions Multiply the inequality by the LCD to eliminate fractions within the inequality (Multiplicative Law of Inequalities) x 3 4x 2 6 < 6 + 2 2 3 3 3x 9 < 8x 3 Simplify Simplify the inequality by combining like terms and dividing by common factors Already simplified. 108
Steps Discussion 4 Decide on direction Think through which direction to isolate the variable Move x to the right side because 8x 3x is more simple Isolate the variable in the inequality Use the Additive Property of Inequalities to create an inequality with all terms of the variable on the same side of the inequality 3x 9 < 8x 3x 9( 3 x) < 8x ( 3 x) 9 < x 6 Continue the Isolation Use the Additive Property of Inequalities to create an inequality with the constants on the other side of the inequality 9 < x 9( 4) < x ( 4) 13 < x 109
Chapter 3 Graphing Fundamentals Steps 7 Complete the isolation Discussion Use the Multiplicative Property of Inequalities to multiply each side by the multiplicative inverse. If the multiplicative inverse is negative, you must reverse the direction of the inequality Multiplicative inverse: 1 1 1 ( 13) < ( x) 13 < x 8 Represent the solution set graphically Identify the end of the line segment, determine if open or closed, and identify direction of the solution set Point is 13, open, and goes to infinity 13 ) 13 0 OR 0 110
Steps 9 Test and Validate solution set Discussion Pick two points from the graph (one in the solution set and one not in the solution set; note: pick easy calculation points) Pick: 0 (in the solution set) and 10 (not in the solution set) 1 1 ( x 3) < (4x + 2) 1 1 ( 0 3) < (4 0 + 2) 3 2 < (yes) 1 1 ( 10 3) < (4( 10) + 2) 13 38 < 6. < 12.66 (no) 111
Chapter 3 Graphing Fundamentals TIPS FOR SUCCESS 3.1 1. Choose a direction of the inequality that minimizes mistakes. 2. Validate the solution set carefully picking simple numbers from the graphical solution for testing. 3. Document carefully each step of the process for consistency. CRITICAL THINKING QUESTIONS 3.1 1. What are the possible variations of symbols used to represent different types of inequalities? 2. What are the components in representing a solution set on a number line? 3. What are the components in representing a solution set in interval notation? 4. How do you represent an end point not in the solution set? a. symbolically b. graphically c. in interval notation. What properties do we use when we solve linear inequalities? 6. What are the critical techniques or steps in the methodology for solving linear equations? 112
7. Which of the techniques or steps from Question 6 can you use in solving linear inequalities? 8. Which steps in the methodology for solving a linear inequality are new or differ from steps in the methodology for solving a linear equation? 9. What is the most common error that occurs when multiplying by the multiplicative inverse of a negative coefficient? 10. What issues can you identify that might make solving linear inequalities difficult for some students? DEMONSTRATE YOUR UNDERSTANDING 3.1 Solve each given inequality for the indicated variable. 1. Solve for x: ( x) + 2 < ( x 2) 3 2 113
Chapter 3 Graphing Fundamentals 2. Solve for a: 3(2a 4) 3 + 2(3a + 7) 3. Solve for x: 3x 4 + x > 7 9x 4. Solve for y: 3 2 y + (4y ) + 2y 11 114
. Solve for x: 2x 4 < 2x + IDENTIFY AND CORRECT THE ERRORS 3.1 In the second column, identify the error you find in each of the following worked solutions and describe the error made. Solve the problem correctly in the third column. Problem Describe Error Correct Process 1. Solve for x: 1 ( ) 2 7 2 x + < Wrong process: Not eliminating the fractions Worked Solution (What is wrong here?) 1 ( x ) + 2 < 7 2 x < x < 10 0 10 11
Chapter 3 Graphing Fundamentals Problem Describe Error Correct Process 2. Solve for x: 6 < 3(x 4) Worked Solution (What is wrong here?) Wrong process: Not testing correctly by visualizing the opposite direction. 6 < 3( x 4) 6 < 3x + 12 6 < 3x 2 > x 0 2 3. Solve for x: 3x 4 7 Wrong process: Eliminating the equal sign Worked Solution (What is wrong here?) 3x 4 7 3x 11 x < 11 3 0 11 3 4. Solve for s: 2s > Worked Solution (What is wrong here?) Wrong process: Negative coefficient not reversing the direction of inequality 2s > 2s > 2 2 s > 2 3 2 2 116
Problem Describe Error Correct Process. Solve for x: 3 + (2x 7) 2x Worked Solution (What is wrong here?) Wrong process: Correct solution, but the check is wrong 3+ (2x 7) 2x 3+ 10x 3 2x 10x 38 2x 8x 38 4 8x 42 42 x 8 21 x 4 0 21 4 Test with 20 and 22: 3+ (2 20 7) 2 20 3+ (43) 44 3+ 21 44 212 44 (yes) 3+ (2x 7) 2x 3+ (2 22 7) 2 22 3+ (34) 48 167 48 (yes) 117