Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Solving Absolute Value INEQUALITIES Section 2.7 Advanced Algebra 1 x 3 An absolute value equation is an equation that contains a variable inside the absolute value sign. This absolute value equation represents the numbers on the number line whose distance from 0 is equal to 3. Two numbers satisfy this equation. Both 3 and -3 are 3 units from 0. Look at the number line and notice the distance from 0 of -3 and 3. 3 units 3 units -3 0 3 Do you remember how to solve this: 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 • Set up two equations to solve. x+2=7 x=5 or x + 2 = -7 x = -9 Solving Absolute Value Inequalities Definition: x > a means x > a or x < -a -a 0 a Definition: x a means x a and x a -a 0 a Keys to Solving Absolute Value Inequalities GreatOR Less ThAND x a x a xa OR x a xa AND x a 2 x3 2 3 2 x3 2 3 OR GreatOR 2 x 3 2 3 2 3 x 3 3 2 3 2 3 x 3 3 2 3 2x 9 6 2x 9 6 2x 3 3 x 2 2x 15 OR 15 x 2 5 2x 4 5 2x 4 2x 1 1 x 2 1 9 <x< 2 2 Less ThAND 5 2x 4 AND 2 x 9 9 x 2 AND -1 0 1 2 3 4 5 Your Turn… Make two cases. Solve the two cases independently. Solve | 2x + 3 | < 6. 2 x 3 6 AND 2 x 3 6 2 x 3 AND 3 x AND 2 2 x 9 9 x 2 Your Turn… Make two cases. Solve the two cases independently. Solve | 2x – 3 | > 5. 2 x 3 5 OR 2 x 3 5 2 x 8 OR x 4 OR 2 x 2 x 1 Example 1: |2x + 1| > 7 2x + 1 > 7 or 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. x > 3 or x < -4 -4 Graph the solution. 3 Example 2: This is an ‘and’ statement. (Less thand). |x -5|< 3 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 Pretend that you are allowed to go within 9 of the speed limit of 65mph without getting a ticket. Write an absolute value inequality that models this situation. |x – 65| < 9 Desired amount Acceptable Range Check Answer: x-65< 9 AND x-65> -9 x<74 AND x >56 56<x<74 If a bag of chips is within .4 oz of 6 oz then it is allowed to go on the market. Write an inequality that models this situation. |x – 6| < .4 Desired amount Acceptable Range Check Answer: x – 6 < .4 AND x – 6 > -.4 x < 6.4 AND x > 5.6 5.6< x < 6.4 In a poll of 100 people, Misty’s approval rating as a dog is 78% with a 3% of error. ticket. Write an absolute value inequality that models this situation. |x – 78| < 3 Desired amount Acceptable Range Check answer: x-78 < 3 AND x-78>-3 x<81 AND x>75 75<x<81 Challenges… HINT: Think midpoint and distance. Find the absolute-value inequality statement that corresponds to the inequality –2 < x < 4. Find the absolute-value inequality statement that corresponds to the inequalities x < 19 or x > 24.