CHAPTER 9 Compound Inequalities

Recall and Vocabulary So far, we studied simple inequalities, but now we will study compound inequalities. A Compound Inequality consists of two inequalities connected by the words “AND” or “OR” 2 < x < 6 is a compound inequality, read as “2 is less than x, and x is less than or equal to 6.” 2 3 1 4 5 6 7 8 9

“AND” Situations Graph all real numbers that are greater than zero and less than or equal to 4. -1 -2 1 2 3 4 5 6 0 < x < 4 “AND” situations have the variable between two numbers, so the graph is shaded between two numbers.

Graph all real numbers that are less than –1 or greater than 2. “OR” Situations Graph all real numbers that are less than –1 or greater than 2. -1 -2 1 2 3 4 5 6 x < -1 or x > 2 OR situations have TWO separate answers and are solved and graphed separately, but on the same number line. The graph of this situation goes in opposite directions.

Solving a Compound Inequality – “AND” When solving “and” inequalities, you isolate the variable between the inequality symbols. IMPORTANT – inverse operations apply to the WHOLE THING (that means both sides!). -2 < 3x – 8 < 10

Graph the solution 2 < x < 6 1 2 3 4 5 6 7 8

Solving Compound Inequalities – “OR” When solving “or” inequalities, you MUST solve AND graph each inequality separately (but on the same number line). 3x + 1 < 4 OR 2x – 5 > 7 3x + 1 < 4 2x – 5 > 7

Graph the solution -3 > x OR x > 2 -4 -3 -5 -2 -1 1 2 3

REMINDER! -2 < -x < 5 2 > x > -5 When you divide or multiply by a negative number, you must reverse BOTH signs if an “AND” situation. In the “OR” inequalities, only reverse it if it applies to the individual inequality. -2 < -x < 5 2 > x > -5

Solve -2 < -2 – x < 1

Graph the solution -3 < x < 0 Examples Graph the solution -3 < x < 0 -3

Writing Compound Inequalities Water is a nonliquid when the temperature is 32 degrees F or below, or is at least 212 degrees F. t < 32 or t > 212 b. A refrigerator is designed to work on an electric line carrying from 115 to 120 volts. 115 < v < 120

RECAP Solve inequalities just as you would equations using inverse operations. KNOW WHEN TO FLIP THE SIGNS!!! Make sure your graph agrees with the inequalities involved. What does the graph of a compound inequality (AND) case look like? OR Case?