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 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 6|5x + 2| = 312 |5x + 2| = 52 Same Opposite Isolate the absolute value expression by dividing by 6. 6|5x + 2| = 312 |5x + 2| = 52 Same Opposite 5x + 2 = 52 5x + 2 = -52 5x = 50 5x = -54 x = 10 or x = -10.8 Check: 6|5x + 2| = 312 6|5x + 2| = 312 6|5(10)+2| = 312 6|5(-10.8)+ 2| = 312 6|52| = 312 6|-52| = 312 312 = 312 312 = 312

3|x + 2| -7 = 14 x + 2 = 7 x + 2 = -7 x = 5 or x = -9 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 Opposite x + 2 = 7 x + 2 = -7 x = 5 or x = -9 Check: 3|x + 2| - 7 = 14 3|x + 2| -7 = 14 3|5 + 2| - 7 = 14 3|-9+ 2| -7 = 14 3|7| - 7 = 14 3|-7| -7 = 14 21 - 7 = 14 21 - 7 = 14 14 = 14 14 = 14

Review of the Steps to Solve a Compound Inequality: Example: 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: 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 you are working with an ‘and’ statement. Remember: “Less thand” If you have a 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: |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7 This is an ‘or’ statement. (Greator). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7 2x + 1 >7 or 2x + 1 <-7 x > 3 or x < -4 3 -4

Example 2: This is an ‘and’ statement. (Less thand). |x -5|< 3 Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. |x -5|< 3 x -5< 3 and x -5< 3 x -5< 3 and x -5> -3 x < 8 and x > 2 2 < x < 8 8 2

Charts to Help Copy the chart from the following page in your notes! Algebra 1 Pg 354 Algebra 2 Pg 53