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Warm Up Algebra 1 Book Pg 352 # 1, 4, 7, 10, 12 Warm Up Answers Algebra 1 Book Pg 352 # 1, 4, 7, 10, 12 1. x < 5 4. x < -10 7. x < 2 10. x > -9 or x < -12 12. x < -1 or x > 29/5 Solving Linear Inequalities Algebra 1 Textbook section 6.4 Algebra 2 Textbook section 1.7 The student will be able to: Solve and graph absolute value equations Solve and graph absolute value inequalities Solving Absolute Value Equations • Absolute value is denoted by bars on each side of a number or expression |3|. • Absolute value represents the distance a number is from 0. Thus, it is always positive. • |8| = 8 and |-8| = 8 Solving absolute value equations • First, isolate the absolute value expression. • Set up two equations to solve. • Same and Opposite • For the first equation, write it the Same way you see it without the absolute value bars and solve. • For the second equation, write it the Opposite way without the absolute value bars and solve • Only write the opposite of the RIGHT SIDE of the equal sign. • Always check the solutions. 6|5x + 2| = 312 • Isolate the absolute value expression by dividing by 6. 6|5x + 2| = 312 |5x + 2| = 52 Same 5x + 2 = 52 5x = 50 x = 10 Opposite or 5x + 2 = -52 5x = -54 x = -10.8 •Check: 6|5x + 2| = 312 6|5(10)+2| = 312 6|52| = 312 312 = 312 6|5x + 2| = 312 6|5(-10.8)+ 2| = 312 6|-52| = 312 312 = 312 3|x + 2| -7 = 14 • Isolate the absolute value expression by adding 7 and dividing by 3. 3|x + 2| -7 = 14 3|x + 2| = 21 |x + 2| = 7 Same x+2=7 x=5 Opposite or •Check: 3|x + 2| - 7 = 14 3|5 + 2| - 7 = 14 3|7| - 7 = 14 21 - 7 = 14 14 = 14 x + 2 = -7 x = -9 3|x + 2| -7 = 14 3|-9+ 2| -7 = 14 3|-7| -7 = 14 21 - 7 = 14 14 = 14 Review of the Steps to Solve a Compound Inequality: ● Example: 2 x 3 2 and 5 x 10 ● You must solve each part of the inequality. ● The graph of the solution of the “and” statement is the intersection of the two inequalities. Both conditions of the inequalities must be met. ●In other words, the solution is wherever the two inequalities overlap. ●If the solution does not overlap, there is no solution. Review of the Steps to Solve a Compound Inequality: ● Example: 3x 15 or -2x +1 0 ● You must solve each part of the inequality. ● The graph of the solution of the “or” statement is the union of the two inequalities. Only one condition of the inequality must be met. ● In other words, the solution will include each of the graphed lines. The graphs can go in opposite directions or towards each other, thus overlapping. ● If the inequalities do overlap, the solution is all real numbers. Solving an Absolute Value Inequality ● ● ● Step 1: Rewrite the inequality as an AND or an OR statement : ● If you have a or you are working with an ‘and’ statement. Remember: “Less thand” ● If you have a or you are working with an ‘or’ statement. Remember: “Greator” Step 2: In the second equation you must negate the right hand side and reverse the direction of the inequality sign. Set up a SAME inequality and an OPPOSITE inequality Solve as a compound inequality. Example 1: ● ● ● ● This is an ‘or’ statement. (Greator). Rewrite. |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7 2x + 1 >7 or 2x + 1 <-7 x > 3 or In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. x < -4 Graph the solution. -4 3 Example 2: ● |x -5|< 3 This is an ‘and’ statement. (Less thand). ● x -5< 3 and x -5< 3 x -5< 3 and x -5> -3 Rewrite. ● ● ● In the 2nd inequality, reverse the inequality sign and negate the right side value. x < 8 and x > 2 2<x<8 Solve each inequality. Graph the solution. 2 8 Charts to Help ● ● ● Copy the chart from the following page in your notes! Algebra 1 Pg 354 Algebra 2 Pg 53