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 and 2.6 are. The arrow shows that the graph extends indefinitely to the _________ 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________

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 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  4x  20 Divide by –4 and reverse the sense of the inequality:  4x   4  20  4 Simplify: x  5 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. 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. 2x  4 Divide by 2: 2x  2  4  2 Simplify: x  2 If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one. 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. 2x  3  7 subtract 3: 2x  3  3  7  3 Simplify: 2x  4. If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one. Addition and Subtraction properties If a  b, then a  c  b  c. If a  b, then a  c  b  c. ExampleThe Property In WordsProperty 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. SolutionWe will collect variable terms on the left and constant terms on the right. 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 4x  5  9x  10 This is the given inequality.

(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. (greatOR) To put together two pieces using interval notation, use the symbol for “union”: These rules are valid if 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  4| < 3 means  3< x  4< 3 |X| < c means  c < X < c 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)