Algebra 2 09/19-20/16 EQ: How do I solve absolute value equations and inequalities? How do I solve compound inequalities HW: pg 154 # 3-35 all (due wed) Bring textbook - sub Quiz corrections

Solve compound inequality 2 types of problems AND 2< 3x -5 < 10 AND 9 < 3x AND x + 8 < 18 OR 2 < 3x - 7 0R 2x > 18

2 ≤ 3x – 4 < 11 2 ≤ 3x – 4 6 ≤ 3x 2 ≤ x 3x – 4 < 11 3x < 15 x < 5

1 2 c ≥ -2 AND 2c + 1 < 1 Solve and Graph

x – 5 < -2 OR -2x≤ -10 Solve and Graph

AND or OR What are the differences and similarities between solving and graphing AND Or inequalities? Similarities: Differences:

Absolute Value | | Absolute value measures the distance from zero. Absolute value is always positive | 5 | = 5 |-5| = 5

Solving equations |3x + 14| = 7 Break into 2 parts Positive 3x + 14 = 7 -14 -14 3x = -7 x = −7 3 Negative 3x + 14 = - 7 -14 -14 3x = - 21 x = -7

|x + 3| - 9 = 5 Simplify outside first by adding 9 to both sides, than separate into 2 parts | x + 3| =14 x + 3 = 14 x + 3 = -14 x = 11 x = -17 POSITIVE SIDE NEGATIVE SIDE

2|x + 5| = 14 Our goal is to get the absolute value by itself so divide both sides by 2. NEVER DISTRIBUTE A NUMBER INSIDE THE ABSOLUTE VALUE!!!!! | x + 5 | = 7 x + 5 = 7 x + 5 = -7 x = 2 x = -12 POSITIVE SIDE NEGATIVE SIDE

|3x – 4| + 6 = 3 Isolate | | (absolute value by subtracting 6) |3x -4| = -3 Recall the definition of absolute value (distance from zero and makes the answer positive). Can we have a negative answer (-3)? No In this case we have NO SOLUTION!

-3|x +5| + 7 = -2 -3|x + 5| + 7 = -2 subtract 7 to both sides -3|x + 5| = -9 divide both sides by -3 |x + 5| = 3 break into 2 problems x + 5 = 3 x + 5 = -3 x = -2 x = -8

Know it. Copy the table of pg151 in your notes Know it!!! Copy the table of pg151 in your notes. Critical Thinking What connections can you make to graphing inequalities? ( Hint: AND / OR graphs)

|2x – 1| > 5 What kind of graph? Is greater than… mORe answers OR graph Split into 2 parts (note what happens on the negative side) 2x -1 > 5 2x -1 < -5 2x > 6 2x < -4 x > 3 x < -2

|3x + 1| ≤ 8 What kind of graph? Is less than… less answers AND graph Split into 2 parts (note what happens on the negative side) 3x -1 ≤ 8 3x -1 ≥ -8 3x ≤ 9 3x ≥ -7 x ≤ 3 x ≥ - 7 3 about -2.33

-4|x + 3| +8 > 4 -4|x + 3| > - 4 subtracted 8 from both sides |x + 3| < 1 divided by -4 (flipped inequality) separate into 2 parts x + 3 < 1 x + 3 > -1 x < -2 x > -4

-4|x + 3| +8 < 12 -4|x + 3| < 4 subtracted 8 from both sides |x + 3| > -1 divided by -4 (flipped inequality) Even though we got a negative number, we can come up with numbers that are bigger than -1 (0, 1, 2, …) x + 3 > - 1 OR x + 3 < 1 x > -4 OR x < -2 ALL REAL NUMBERS

-4|x + 3| +8 > 12 -4|x + 3| > 4 subtracted 8 from both sides |x + 3| < -1 divided by -4 (flipped inequality) Can the absolute value produce a value that is less than a negative number? No - therefore, there is no solution