1. Previously you studied four types of simple inequalities. In this lesson you will study compound inequalities. A compound inequality consists of two inequalities connected by and or or. Solving Compound Inequalities Write an inequality that represents the set of numbers and graph the inequality. All real numbers that are greater than zero and less than or equal to 4. SOLUTION 0 < x  4 This inequality is also written as 0 < x and x  4. 401235–1

2. Previously you studied four types of simple inequalities. In this lesson you will study compound inequalities. A compound inequality consists of two inequalities connected by and or or. Solving Compound Inequalities Write an inequality that represents the set of numbers and graph the inequality. All real numbers that are greater than zero and less than or equal to 4. All real numbers that are less than –1 or greater than 2. SOLUTION 0 < x  4 This inequality is also written as 0 < x and x  4. x 2 401235–13 0124–2 SOLUTION

3. Write an inequality that describes the elevations of the regions of Mount Rainier. Compound Inequalities in Real Life Timber region below 6000 ft SOLUTION Let y represent the approximate elevation (in feet). Timber region: 2000  y < 6000 Alpine meadow region Timber region Glacier and permanent snow field region 14,140 ft 7500 ft 6000 ft 2000 ft 0 ft

4. Write an inequality that describes the elevations of the regions of Mount Rainier. Compound Inequalities in Real Life Timber region below 6000 ft SOLUTION Let y represent the approximate elevation (in feet). Timber region: 2000  y < 6000 Alpine meadow region Timber region Glacier and permanent snow field region 14,140 ft 7500 ft 6000 ft 2000 ft 0 ft Alpine meadow region below 7500 ft Alpine meadow region: 6000  y < 7500

5. Write an inequality that describes the elevations of the regions of Mount Rainier. Compound Inequalities in Real Life Timber region below 6000 ft SOLUTION Let y represent the approximate elevation (in feet). Timber region: 2000  y < 6000 Alpine meadow region Timber region Glacier and permanent snow field region 14,140 ft 7500 ft 6000 ft 2000 ft 0 ft Alpine meadow region below 7500 ft Alpine meadow region: 6000  y < 7500 Glacier and permanent snow field region Glacier and permanent snow field region: 7500  y  14,410

6. Solve –2  3x – 8  10. Graph the solution. Solving a Compound Inequality with And SOLUTION Isolate the variable x between the two inequality symbols. –2  3 x – 8  10 Write original inequality. 6  3 x  18 Add 8 to each expression. 2  x  6 Divide each expression by 3. The solution is all real numbers that are greater than or equal to 2 and less than or equal to 6. 40123657

7. Solve 3x + 1 7. Graph the solution. Solving a Compound Inequality with Or SOLUTION A solution of this inequality is a solution of either of its simple parts. You can solve each part separately. The solution is all real numbers that are less than 1 or greater than 6. 3x + 1 < 42x – 5 > 7 or 3x < 32x > 12 or x < 1x > 6 or 4012365–17

8. Solve –2 < –2 – x < 1. Graph the solution. Reversing Both Inequality Symbols SOLUTION Isolate the variable x between the two inequality signs. –2 < –2 – x < 1 0 < –x < 3 0 > x > –3 To match the order of numbers on a number line, this compound is usually written as –3 < x < 0. The solution is all real numbers that are greater than –3 and less than 0. Write original inequality. Add 2 to each expression. Multiply each expression by –1 and reverse both inequality symbols. 0– 4– 4–3–2–121–53

9. You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the smallest value d might have? Modeling Real-Life Problems DRAW A DIAGRAM A good way to begin this problem is to draw a diagram with the school at the center of a circle. Problem Solving Strategy SOLUTION Bill’s home is somewhere on the circle with radius 3 miles and center at the school. Mary’s home is somewhere on the circle with radius 2 miles and center at the school.

10. You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the smallest value d might have? Modeling Real-Life Problems If both homes are on the same line going toward school, the distance is 1 mile. SOLUTION

11. You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the largest value d might have? Modeling Real-Life Problems SOLUTION If both homes are on the same line but in opposite directions from school, the distance is 5 miles.

12. You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the largest value d might have? Modeling Real-Life Problems SOLUTION If both homes are on the same line but in opposite directions from school, the distance is 5 miles. Write an inequality that describes all the possible values that d might have. The values of d can be described by the inequality 1  d  5. SOLUTION