1 of 14 Algebra II A – Guided Notes Name ________________________________ 1-­‐6 Guided Notes Period ______________ Solving Compound and Absolute Value Inequalities Learning Matrix Goal #13: I can solve compound inequalities. Learning Matrix Goal #14: I can graph the solution set of compound inequalities. Learning Matrix Goal #15: I can solve absolute value inequalities. Learning Matrix Goal #16: I can graph the solution of absolute value inequalities. How are compound inequalities used in medicine? Read this information on page 40 of your textbook and explain here. If you are scheduled to have a glucose tolerance test at 10 A.M., at what hour should you begin fasting? Explain Medicine: What does a glucose tolerance test measure? Define in your own words: Compound inequality-­‐ Intersection-­‐ 2 of 14 Key Concept “And” Compound Inequality Words: Example: Another way of writing x ≥ -­‐1 and x < 2 is _________________________________. Both forms are read ________________________________________________________________________. (This is the way we will write our final answers.) Now take notes from Khan Academy. Go to Math Then Algebra Then Linear Inequalities Then Compound and Absolute Value Inequalities Then watch the video called Compound Inequalities 3 (that exact title) -­‐16 ≤ 3x + 5 ≤ 20 Then watch the video called Compound Inequalities (that exact title – the first one listed) -­‐5 ≤ x – 4 ≤ 13 3 of 14 -­‐12 < 2 – 5x ≤ 7 (3:55) (Stop at 8:35) Study Tip Interval Notation The compound inequality -­‐1 ≤ x < 2 can be written as __________, indicating that the solution set is the set of all numbers ________________ -­‐1 and 2, _______________ -­‐1, but _______________ 2.
Example 1 Solve an “and” Compound Inequality Copy example on p.40. Solve 13 < 2x + 7 ≤ 17. Graph the solution set on a number line. You may use either Method given. If you are not sure about them, ask to do the problems with the teacher. 4 of 14 Your Turn Solve 10 ≤ 3y -­‐2 < 19. Graph the solution set on a number line. Answer in both set and interval notation. Now take notes from Khan Academy. Go to Math Then Algebra Then Linear Inequalities Then Compound and Absolute Value Inequalities Then watch the video called Compound Inequalities 4 (that exact title) 5x – 3 < 12 and 4x + 1 > 25 What is the solution to this problem? Why? Define in your own words: Union-­‐ 5 of 14 Key Concept “Or” compound Inequalities Words: Example: Study Tip Interval Notation In interval notation, the symbol for the union of the two sets is ∪. The compound inequality y > -­‐1 or y ≤ -­‐7 is written as (-­‐∞, -­‐7] ∪ (-­‐1, +∞) indicating that all values less than and including -­‐7 are part of the solution set. In addition, all values greater than -­‐1, not including -­‐1, are part of the solution set.
To Help You Remember: As a memory device, the word or begins with the letter o which is found in the worked union, while and begins with the letter a witch is not found in union. Example 2 Solve an “Or” Compound Inequality Solve y – 2 > -­‐3 or y + 4 ≤ -­‐3 Graph the solution set on a number line. (You solve each problem on its own and then graph them on the same number line) Your Turn Solve x + 3 < 2 or –x ≤ -­‐4. Graph the solution set on a number line. Answer in both set and interval notation. 6 of 14 Now take notes from Khan Academy. Go to Math Then Algebra Then Linear Inequalities Then Compound and Absolute Value Inequalities Then watch the video called Absolute Value Inequalities (that exact title) Take very detailed notes and pay close attention to what he is saying, especially at the very beginning. Copy down these notes first. |x| < 12 After you copy the above notes, listen to what he says (read the subtitles if needed) and fill in the blanks. We want all the numbers that are _________________________________ 12 away from 0. You can go all the way to ___________________ and you can go all the way to ________________________ Anything that is ______________________________ those two numbers is going to have an absolute value of less than 12. |7x| ≥ 21 (2:20) After you copy the above notes, listen to what he says (read the subtitles if needed) and fill in the blanks. Whatever is inside our absolute value sign is 21 or more away from 0. So we want all the numbers that are a distance from 0 greater than or equal to 21. So 7x needs to be equal to numbers that are __________________ -­‐21 or _____________________ 21 (Stop at 6:08) Example 3 Solve an Absolute Value Inequality (<) Solve | a | < 4. Graph the solution set on a number line. Copy example on p.42. 7 of 14 Notice that the graph of |a| < 4 is the same as the graph of _________________ and _____________________ Therefore the solution set is _____________________________ Your Turn Solve 3 > |d| Graph the solution set on an number line. Answer in both set and interval notation. Example 4 Solve an Absolute Value Inequality (>) Copy example on p. 42 Study Tip Absolute Value Solve |a| > 4. Graph the solution set on a number line. Inequalities Because the absolute value of a number is never __________________, *the solution of an inequality like |a| < -­‐4 is the ____________________ (which means “no solution”. *the solution of an Notice that the graph of |a| > 4 is the same as the graph of inequality like _________________ and _____________________ |a|> -­‐4 is the set of ______________________. Therefore the solution set is _____________________________ The absolute value of a number is always positive. A positive number is always greater than a negative. 8 of 14 Your Turn Solve 3 < |d|. Graph the solution set on a number line. Answer in both set and interval notation. _______________________________________________________________________________________________________________ PARTNER ACTIVITY – Find a partner who is at this place in the notes. Work on the partner activity. You will be creating a number line on the floor using tape, the floor tiles, and string. You will need to know each block is a foot wide. • Tape the end of the string where two floor tiles meet. • You are going to graph |x| < 5. • You want all numbers that are < 5 away from 0 • Stretch the string out 5 blocks o Describe what numbers are within that distance. o Now write you’re your above answer as an inequality. • Now in the opposite direction, stretch the string the same distance o Describe what numbers are within that distance. o Now write your above answer as an inequality. • Based on what you observed, describe what the solution to |x| < 5 would be. • Now write an inequality for |x| < 5 • DEBREIF WITH THE TEACHER AND SUMMERIZE WHAT WE TALK ABOUT HERE • NOW you are going to graph |x| > 3. • You want all numbers that are > 3 away from 0 • Stretch the string out 3 blocks 9 of 14 o Now write your above answer as an inequality. •
Now in the opposite direction, stretch the string the same distance o Describe what numbers that are beyond that distance. o Now write your above answer as an inequality. o Describe what numbers that are beyond that distance. •
Based what you observed, describe what the solution to |x| < 3 would be. •
Now write an inequality for |x| < 3 DEBREIF WITH THE TEACHER AND SUMMERIZE WHAT WE TALK ABOUT HERE ______________________________________________________________________________________________________________ Key Concept Absolute Value Inequalities Symbols: Examples: Now take notes from Khan Academy at video we last watched Go to Math Then Algebra Then Linear Inequalities Then Compound and Absolute Value Inequalities Then watch the video called Absolute Value Inequalities (that exact title) •
10 of 14 (start at 6:10) |5x + 3| < 7 (8:20) Write down the rule that is given, then rewind and listen to what he is saying. (10:25) 2!
5
+ 9 > 7
7
Example 5 Solve a Multi-­‐Step Absolute Value Inequality Solve |3x -­‐12| ≥ 6. Graph the solution set on a number line. Follow the example on page 42. 11 of 14 Your Turn Solve |2x – 2| ≥ 4. Graph the solution set on a number line. Answer in both set and interval notation. Now take notes from Khan Academy at video we last watched Go to Math Then Algebra Then Linear Inequalities Then Compound and Absolute Value Inequalities Then watch the video called Absolute Value Inequalities Example 2 (that exact title) |p – 12| + 4 < 14 Now take notes from Khan Academy at video we last watched Go to Math 12 of 14 Then Algebra Then Linear Inequalities Then Compound and Absolute Value Inequalities Then watch the video called Absolute Value Inequalities Example 3 (that exact title) |y| + 22 ≤ 13.5 SPECIAL CASE!!! Your Turn Solve |x + 2| +3 < 0. Graph the solution set on a number line. Answer in both set and interval notation. SPECIAL CASE!!!! Your Turn Solve |x + 2| +3 > 0. Graph the solution set on a number line. Answer in both set and interval notation. 13 of 14 Example 6 Write an Absolute Value Inequality textbook page 43 Job Hunting. To prepare for a job interview, Megan researches the position’s requirements and pay. She discovers that the average starting salary for the position is $38,500, but her actual starting salary could differ from the average by as much as $2450. a. Write an absolute value inequality to describe this situation. b. Solve the inequality to find the range of Megan’s starting salary. REVIEW IN YOUR OWN WORDS Name four different ways to show the solution of an inequality. 1. 2. 3. 4. How do we know when to reverse the inequality sign? How do we know when to use an open circle in a line graph? Closed circle? Do the following: |3x + 5| ≤ -­‐2 14 of 14 |x + 3| ≥ -­‐6 Now complete page 43 #1-­‐14, 15-­‐23 odd, 27-­‐45 odd, 46, 47, 53. Be sure to answer all problems in both set and interval notation. When assignment is complete, you should check your solutions (get them from the solutions folder). Mark correct problems with a star. Mark incorrect problems with an X and them make corrections on the problems that you had incorrect.