| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
Here are a few recommended readings before getting started with this lesson.
Vincenzo is getting ready to drive home from vacation at the beach. He is sure that if he drives at 50 miles per hour, he will be home in 40 to 55 minutes.
Combining two or more inequalities with the word and
or or
yields what is called a compound inequality.
Compound Inequality | Is Read As |
---|---|
x<5 or x>8 | x is less than 5 or greater than 8. |
x>2 and x≤4 | x is greater than 2 and less than or equal to 4. |
andare commonly written without using this word. Consider the following example.
x is greater than 2is equivalent to
2 is less than x.With this change, the inequality can be rewritten as follows.
and.
LHS+1<RHS+1
Add terms
LHS/2<RHS/2
Simplify quotient
LHS+1≤RHS+1
Add terms
LHS/2≤RHS/2
Simplify quotient
andinequality. Therefore, the two solution sets can written separately with the word
and,or they can be combined as follows.
or,a solution of either individual inequality is a solution of the compound inequality. Consider the following compound inequality.
and,however, must be a solution of both individual inequalities. Consider the following inequality.
or,compound inequalities written with
anddo not always extend infinitely.
Kevin's class is having a work experience internship this week. He is working with a farmer to test the speed of an autonomous tractor in a field.
In the first test, the tractor started from a point 2 miles away from the barn. After a half hour, the tractor was at least 10 miles away from the barn. In the next test, the tractor started at a point 1.5 miles away from the barn. When it stopped 45 minutes later, it was less than 20 miles from the barn.
and.
andcan be rewritten as follows.
On this case, since the inequality is strict, the solution set of the inequality r<24.7 is made of the numbers to the left of 24.7. Since 24.7 is not included in the solution set, an open circle is used instead.
Since the compound inequality is written with and,
its solution set is made of the numbers that satisfy both inequalities.
Ignacio is interning at the local disaster preparedness center. He is using a simulator to help to secure the unsafe area near a volcano that is about to erupt. He is in charge of marking safe distances to the east and the west of the volcano. He initially marked the safe distance with flags, one 30 miles to the east of the base of the volcano and the other 15 miles to the west.
As time passes and data comes in, Ignacio realizes that his estimates were wrong. He notes that the flag to the west is less than half the distance away from the volcano that it should be. On the other hand, he calculates that the eastern flag covers less than two thirds of the actual necessary safe distance from the volcano.
LHS⋅23<RHS⋅23
ba⋅ab=1
ca⋅b=ca⋅b
Multiply
Calculate quotient
Rearrange inequality
or.
Similarly, the graph of the solution set of d>45 is made of every number to the right of 45, not including 45.
Finally, since it is written with or,
the graph of the compound inequality is the combination of both solution sets and does not need to be adjusted or limited.
Tearrik is spending his week of work experience at a local bakery. On his first day, he bought 100 cookies at a discount. He decides to eat 5 cookies a day until they are all gone. Tearrik is also allowed to bring home 3 cookies per day, which he gives to his brother. Tearrik's brother decides that he will not eat his cookies until he has at least 90 saved up.
and.
Similarly, the solution set of the inequality d≥30 is made of the numbers greater than or equal to 30. Since thi inequality is not strict, the circle is closed.
The solution set of the compound inequality is made of the numbers that satisfy both inequalities at the same time. Since there are no such numbers, the compound inequality has no solution.
This confirms that the brothers will not eat cookies together.
For his internship, Davontay is working in a research lab. He is researching what tools are used to measure really low and really high temperatures. He found out that a thermocouple thermometer can measure temperatures lower than 3272∘ F, while a pyrometer thermometer can measure temperatures greater than or equal to 973 K.
To help find these temperatures in degrees Celsius, the relationships between the different temperature scales are shown in the following table. It should be noted that C refers to a temperature in degrees Celsius, F is the temperature in degrees Fahrenheit, and K refers to kelvins.
Fahrenheit | Kelvin |
---|---|
59C+32=F | C+273=K |
or.
LHS−32<RHS−32
LHS⋅95<RHS⋅95
ca⋅b=ca⋅b
Multiply
Calculate quotient
Similarly, the solution set of the inequality C≥700 is made of every point to the right of 700, including 700.
Since the inequality is written with or,
the solution set of the compound inequality is made of the numbers that satisfy either inequality. By combining the graphs it can be noted that every number is a solution to the compound inequality.
This means that any temperature can be measured using either of the thermometers.
Diego is helping at a local car dealership for his work experience. After spending time at the dealership, he decides to start saving money to buy a car. His father told him that he would double the amount of money that Diego saves, starting from now. Also, Diego will receive extra 500 dollars from his uncle to help buy the car when he finishes saving.
When looking for prices, Diego notices that most of the cars he likes range from 15 thousand dollars to 18 thousand dollars.