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1.7 Linear Inequalities and Absolute Value Inequalities

1.7 Linear Inequalities and Absolute Value Inequalities. (Rvw.) Graphs of Inequalities; Interval Notation.

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1.7 Linear Inequalities and Absolute Value Inequalities

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  1. 1.7 Linear Inequalities andAbsolute Value Inequalities

  2. (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.

  3. 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.

  4. Graph the solutions of 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 Text Example Note: If the variable is on the left, the inequality symbol shows the shape of the end of the arrow in the graph.

  5. 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 Text Example

  6. 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.

  7. Property The Property In Words Example Addition and Subtraction properties If a<b, then a+c<b+c. If a<b, then a-c<b-c. If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one. 2x+ 3 < 7 subtract 3: 2x+ 3 - 3< 7 - 3 Simplify: 2x< 4. Positive Multiplication and Division Properties If a<b and c is positive, then ac<bc. If a<b and c is positive, then ac<bc. If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one. 2x< 4 Divide by 2: 2x 2< 4  2 Simplify: x< 2 Negative Multiplication and Division Properties If a<b and c is negative, then ac>bc. If a<b and c is negative, then ac>bc. if we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the result is an equivalent inequality. -4x< 20 Divide by –4 and reverse the sense of the inequality: -4x-4 > 20  -4 Simplify: x> -5 (Rvw.) Properties of Inequalities Bottom line: treat just like a linear EQUALITY, EXCEPT you flip the inequality sign if:

  8. 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

  9. If X is an algebraic expression and c is a positive number: 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 . (Rvw) Solving an Absolute Value Inequality *** IMPORTANT: You MUST ISOLATE the absolute value before applying these principles and dropping the bars.

  10. |X| < c means -c < X < c |x- 4| < 3 means -3< x- 4< 3 Text Example (Don’t look at notes, no need to write.) Solve and graph: |x- 4| < 3. Solution 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)

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