Inequalities in One Variable

 Drink at least six glasses of water a day  Store milk at a temperature below 40 degrees F  Eat snacks fewer than 20 calories  Spend at least $10 for a gift  All are examples of inequalities in everyday life.

 You will analyze situations involving inequalities in one variable to find and graph their solutions.

An inequality is a statement that one quantity is less than or greater than another. Less than<Less than or equal to Greater than >Greater than or equal to

Everyday phraseTranslationInequality At least six glassesThe number of glasses is greater than or equal to 6 Below 40 degreesThe temperature is less than 40 o Fewer than 20 caloriesThe number of calories is less than 20 At most $10The price of the gift is less than or equal to $10 Between 35 o and 120 o 35 o is less than the temperature and the temperature is less than 120 o

 You will act out operations on a number line.  Choose ◦ An announcer ◦ A recorder ◦ Two walkers  The two walkers make a number line from -10 to 10 on the floor.  The announcer and recorder use the Toe the Line table.

 The announcer calls out the operation for Walker A and B.  The walkers perform the operations on their numbers by walking to the resulting value on the number line.  The recorded longs the position of each walker after each operation.  For a trial act out the first operation in the table. ◦ Walker A should stand at position 2 and Walker B should be at position 4. ◦ Enter the inequality sign that describes the relative position of walker A and B on the number line. (<)

 Call out the operations.  After the walkers calculate their new position, record their new positions on the table  Discuss together which inequality sign should be placed between the positions.

 What happens to the walkers’ relative positions on the number line ◦ when the operation adds or subtracts a positive number? ◦ Adds or subtracts a negative number? ◦ Does anything happen to the direction of the inequality symbol?  What happens to the walkers’ relative positions on the number line when the operation multiplies or divides by a positive number? ◦ Does anything happen to the direction of the inequality symbol?

 What happens to the walkers’ relative positions on the number line when the operation multiplies or divides by a negative number? ◦ Does anything happen to the direction of the inequality symbol?  Which operation on an inequality reverse the inequality symbol?  Does it make any difference which numbers you use? (Fractions, decimals, and integers)  Check your findings about the effects of adding, subtracting, multiplying, and dividing by the same number on both sides of an inequality by creating your own table of operations and walker positions.

 Erin says, “I lose 15 minutes of sleep every time the dog barks. Last night I got less than 5 hours of sleep. I usually sleep 8 hours.” Find the number of times Erin woke up. Let x = the number of times Erin woke up. The number of hours Erin slept is 8 hours, minus 15 min (1/4 hour) times x (the number of times she woke up). This total is less than 5 hours.

Subtracting 5 from both sides Simplifying Dividing both sides by The dog woke her up more than 12 times. However, Erin can only wake up a whole number of times so the solution might be more accurately written as x>12, where x is whole number.

 In example B, the inequality x<5 was written to represent the situation where Erin slept less than 5 hours, and her sleep time was 8 hours minus 0.25 hour for each time the dog barked. However, Erin can’t sleep less than 0 hours, so a more accurate statement would be the compound inequality 0≤8-0.25x≤5. You can solve a compound inequality in the same way you solved other inequalities; you just need to make sure you do the same operation to all three parts. Solve this inequality for x and graph the solution. Subtracting 5 from all three parts Simplifying Dividing all three parts by -0.25

 You learned to write and solve one-variable inequalities and interpret the results based on real-world situations.  You graphed solutions to one-variable inequalities on a number line.  You interpreted an interval graphed on a number line as an inequality statement.  You learned the sign-changed rule for multiplying and dividing both sides of a one- variable inequality by a negative number.