Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 1 Linear, Compound, and Absolute Value Inequalities

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1 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 1 What is the following symbol? < The inequality symbols < > are used to compare two real numbers. The meaning of anyone of them depends on the direction in which you read it from left to right or right to left. Example 1: 1 < 2 ( is true) From left to right, you read this, One is less than two. This is a true statement. From right to left, you read this, Two is greater than one which is just as true. Example 2: 5 > 7 ( is false) From left to right, you read this, Five is greater than seven. This is a false statement. From right to left, you read this, Seven is less than five which is just as false. From Example 1 and Example 2, we see that an inequality symbol s meaning depends on the direction in which you read the inequality statement. However, we also see that it is sufficient to read the inequality statement in just one direction in order to determine if the statement is true or if it is false. Example 3: 5 1 From left to right, you read this, Negative five is greater than or equal to one. This is a statement, because -5 is 1. Example 4: 5-6 From left to right, you read this, Negative five is greater than or equal to negative six. This is a statement, because -5 is -6. Example 5: 8 8 From left to right, you read this, Eight is less than or equal to eight. This is a statement, because 8 is 8.

2 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 2 Exercise 1: Plot -5 and 2 on the number line, and use < or > to make a true statement What makes -5 less than 2? One-Dimensional Algebraic Inequalities If we introduce the variable x into an inequality statement, we get an algebraic inequality. Exercise 2a: 2 < x Reading from the variable to the other side, we read this statement all real numbers x that are greater than 2. Graph the inequality, and write the inequality in interval notation. Interval Notation Exercise 2b: x 2 We read this statement all real numbers x that are greater than or equal to 2. Graph the inequality, and write the inequality in interval notation. Interval Notation Compound Inequalities Exercise 3: -1 < x < 5 We read this statement all real numbers x that are greater than -1 and less than 5. Write this compound inequality with the appropriate Boolean logical operator ( and or or ). Graph the inequality, and write the inequality in interval notation. Interval Notation

3 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 3 Exercise 4: [0, 3) Graph the interval, and write it as a compound inequality. Compound Inequality Write the interval with the appropriate Boolean logical operator ( and or or ): Exercise 5: Write the graphed interval using interval notation and inequality notation. ( Interval Notation: Inequality Notation: Exercise 6: Write the graphed interval using interval notation and inequality notation. ] Interval Notation: Inequality Notation: Exercise 7: Write the graphed interval using interval notation and inequality notation. [ ) Interval Notation: Inequality Notation:

4 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 4 Exercise 8a: Solve the compound inequality. Graph the solution set, and write the solution set in interval notation and as. x 5 and x 1 Exercise 8b: Solve the compound inequality. Graph the solution set, and write the solution set in interval notation. x 5 or x 1 Exercise 9a: Solve the compound inequality. Graph the solution set, and write the solution set in interval notation. x 7 and x 2 Exercise 9b: Solve the compound inequality. Graph the solution set, and write the solution set in interval notation. x 7 or x 2

5 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 5 Solving Linear Inequalities in One Variable Exercise 10a: Determine whether or not x = 3 is a solution of the inequality 2x Is 2(3) 1 11 a true statement? Thus, x = 3 a solution of 2x Exercise 10b: Determine whether or not x = 8 is a solution of the inequality 2x Is 2(8) 1 11 a true statement? Thus, x = 8 a solution of 2x Exercise 10c: Determine all solutions of the inequality 2x We cannot test, as we did in Exercises 10a and 10b, each real number in the inequality one by one. Instead, we ll solve the inequality algebraically. 2x 1 11

6 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 6 Exercise 11: Solve the inequality 3x 1 16, and graph the solution set using interval notation. In this exercise, we will see an important algebraic rule for solving inequalities: If you multiply or divide both sides of an inequality statement by a negative number, you must reverse the direction of the inequality. -3x Subtract 1 from each side 1 1-3x 15 Divide each side by -3-3x Reverse the inequality x -5 Exercise 12: Solve the inequality 8 4(2 x) -2x.

7 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 7 Exercise 13 (Section 1.7 #18): Solve the inequality y 3 3y y 3 3y Exercise 14 (Section 1.7 #38): Solve the compound inequality 6 3x 9 0.

8 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 8 Solving Absolute Value Inequalities in One Variable Exercise 15: Solve the absolute value inequality 3x 15, and graph the solution set. Recall that 3 x is either 3x or -3x. We solve 3x < 15 and -3x < 15. It is equivalent to solve this type of absolute value inequality by solving the compound inequality below. -15 < 3x < 15 Theorem: If a is any positive number and if u is any algebraic expression, then u a is equivalent to -a < u < a u a is equivalent to -a u a Exercise 16 (Section 1.7 #48): Solve the absolute value inequality 2 7 y 1 17.

9 Bob Brown, CCBC Essex Math 163 College Algebra, Chapter 1 Section 7 COMPLETED 9 Exercise 17: Solve the absolute value inequality 2x 6, and graph the solution set. Recall that 2 x is either 2x or -2x. We solve 2x > 6 or -2x > 6. It is equivalent to solve this type of absolute value inequality by solving 2x < -6 or 2x > 6 Theorem: If a is any positive number and if u is any algebraic expression, then u a is equivalent to u < -a or u > a u a is equivalent to u -a or u a Exercise 18 (Section 1.7 #58): Solve the absolute value inequality 15 2d 3 6.