Document Sample

1.7 Linear Inequalities and Absolute Value Inequalities (Rvw.) Graphs of Inequalities; Interval Notation There are infinitely many solutions to the inequality x > -4, namely all real numbers that are greater than -4. Although we cannot list all the solutions, we can make a drawing on a number line that represents these solutions. Such a drawing is called the graph of the inequality. Graphs of Inequalities; Interval Notation • Graphs of solutions to linear inequalities are shown on a number line by shading all points representing numbers that are solutions. Parentheses indicate endpoints that are not solutions. Square brackets indicate endpoints that are solutions. (baby face & block headed old man drawings) (Also see p 165 for more clarification.) Do p 176 #110. Emphasize set builder, interval, and graphical solutions. Text Example Graph the solutions of a. x < 3 b. x -1 c. -1< x 3. Solution: a. The solutions of x < 3 are all real numbers that are _________ than 3. They are graphed on a number line by shading all points to the _______ of 3. The parenthesis at 3 indicates that 3 is NOT a solution, but numbers such as 2.9999 and 2.6 are. The arrow shows that the graph extends indefinitely to the _________. -5 -4 -3 -2 -1 0 1 2 3 Note: If the variable is on the left, the inequality symbol shows the shape of the end of the arrow in the graph. Text Example Graph the solutions of a. x < 3 b. x -1 c. -1< x 3. Solution: b. The solutions of x -1 are all real numbers that are _________ than or ___________ -1. We shade all points to the ________ of -1 and the point for -1 itself. The __________ at -1 shows that 1ISa solution for the given inequality. The arrow shows that the graph extends indefinitely to the________. -5 -4 -3 -2 -1 0 1 2 3 Ex. Con’t. Graph the solutions of c. -1< x 3. Solution: c. The inequality -1< x 3 is read "-1 is ______ than x and x is less than or equal to 3," or "x is _________ than -1 and less than or equal to 3." The solutions of -1< x 3 are all real numbers between -1 and 3, not including -1 but including 3. The parenthesis at -1 indicates that -1 is not a solution. By contrast, the bracket at 3 shows that 3 is a solution. Shading indicates the other solutions. -5 -4 -3 -2 -1 0 1 2 3 Note: it must make sense in the original inequality if you take out variable. In this case, does -1 < 3 make sense? If not, no solution. (Rvw.) Properties of Inequalities Property The Property In Words Example Addition and Subtraction If the same quantity is added to or 2x + 3 < 7 properties subtracted from both sides of an subtract 3: If a < b, then a + c < b + c. inequality, the resulting inequality 2x + 3 - 3 < 7 - 3 If a < b, then a - c < b - c. is equivalent to the original one. Simplify: 2x < 4. Positive Multiplication If we multiply or divide both sides 2x < 4 and Division Properties of an inequality by the same Divide by 2: If a < b and c is positive, positive quantity, the resulting 2x 2 < 4 2 then ac < bc. inequality is equivalent to the Simplify: x < 2 If a < b and c is positive, original one. then a c < b c. Negative Multiplication if we multiply or divide both sides -4x < 20 and Division Properties of an inequality by the same Divide by –4 and If a < b and c is negative, negative quantity and reverse the reverse the sense of then ac bc. direction of the inequality symbol, the inequality: If a < b and c is negative, the result is an equivalent -4x -4 20 -4 then a c b c. inequality. Simplify: x -5 Bottom line: treat just like a linear EQUALITY, EXCEPT you flip the inequality sign if: Ex: Solve and graph the solution set on a number line: 4x + 5 9x - 10. Solution We will collect variable terms on the left and constant terms on the right. 4x + 5 9x - 10 This is the given inequality. The solution set consists of all real numbers that are _________ than or equal to _____, expressed in interval notation as ___________. The graph of the solution set is shown as follows: Do p 175#58, 122 (Rvw) Solving an Absolute Value Inequality If X is an algebraic expression and c is a positive number: 1. The solutions of |X| < c are the numbers that satisfy -c < X < c. (less thAND) 2. The solutions of |X| > c are the numbers that satisfy X < -c or X > c. (greatOR) To put together two pieces using interval notation, use the symbol for “union”: These rules are valid if < is replaced by and > is replaced by . *** IMPORTANT: You MUST ISOLATE the absolute value before applying these principles and dropping the bars. Text Example (Don’t look at notes, no need to write.) Solve and graph: |x - 4| < 3. Solution |X| < c means -c < X < c |x - 4| < 3 means -3< x - 4< 3 We solve the compound inequality by adding 4 to all three parts. -3 < x - 4 < 3 -3 + 4 < x - 4 + 4 < 3 + 4 1< x<7 The solution set is all real numbers greater than 1 and less than 7, denoted by {x| 1 < x < 7} or (1, 7). The graph of the solution set is shown as follows: (do p 175 # 72, |x + 1| < -2, | x + 1 | > -2)

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 4 |

posted: | 10/1/2012 |

language: | English |

pages: | 10 |

OTHER DOCS BY dfhdhdhdhjr

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.