02.06 Compound Inequalities
Essential Questions
· How can you create inequalities in one variable and use them to solve problems?
· How can you represent constraints by inequalities?
· How can you interpret solutions as viable or nonviable options in a modeling context?
Absolute Value Inequalities:
Absolute Value Inequalities are problems that involve ranges.
For example: On public stairs, handrails must be installed. The height of the handrails must be within a 3 inch range of 35 inches.
Compound inequality Key words
“and” / “or”
Scenario 1
A fish has to measure between 18 and 24 inches in length.
b>= 18 and b<=24
Writing the compound inequality like this makes it easier to understand that the solutions for b must fall between 18 and 24 inches
Scenario 2
The fish would be either less than 18inches or greater than 24 inches
b<18 or b>24
Compound Inequalities
Compound inequalities are inequalities joined by the word "and" or "or." These two types of compound inequalities have different looking graphs.
Conjunctions
Consider inequalities joined by the word "and." These inequalities have a special name; they are called conjunctions.
The solution to a conjunction is any number that makes both inequalities true.
Graph the conjunction: x > 3 and x < 6.
Notice the word "and" between the inequalities. The solutions for x must fit both conditions stated; the solutions for x must be greater than 3 and less than 6.
Disconjunctions
Now, consider inequalities joined by the word "or."
These inequalities have a special name also; they are called disjunctions. The solution to a disjunction is any number that satisfies either inequality.
Graph the disjunction: x < 3 or x > 6.
By using the word “or” between the inequalities, this disjunction is telling you that solutions for x can fit either of the restrictions: x can be any number less than 3 or any number greater than 6.
Conjunctions
Remember, to graph a compound inequality, you can graph each inequality separately. The part in which they intersect (or overlap) is the solution to the conjunction.
Example 1
−3<x+2≤7.
Method 1 : Separate First
-3 < x+2 <= 7
-3<x + 2 and x + 2 <= 7
-2 -2
-5< x and x< 5
-5<x<5
Method 2 : Leave Together
-3<x + 2 <=7
-2 -2 -2
-5< x < 5
Writing a Conjunction
You will be given the graph of an inequality and asked to come up with the conjunction.
Can you write the conjunction that corresponds with this graph?
Solving a Disjunction
Solve and graph the disjunction: 6 + p < 4 or −5p ≤−30.
There is really only one method to use when solving disjunctions: solve each inequality separately.
Note that the operations needed to isolate the variable may not be the same for both inequalities.
6 / + / p / 4 / or / −5p / ≤ / −30−6 / −6 / or /
−5
/ ≤ /
−5
p / −2 / or / p / ≥ / 6
This solution says that p is less than −2 or greater than or equal to 6.
Let’s graph out disjunction p <-2 or p>=6
Writing a Disjunction
Example 3: Can you write the inequality that corresponds with this graph?
Absolute Value Inequalities
Less than Absolute Value Inequalities
If the absolute value inequality contains a less than symbol (<, <=).
Write two separate inequalities to solve.
1. Drop the absolute value bars.
2. Create a second inequality with the flipped inequality symbol, opposite value, and the word “and” in between.
The graph is between two numbers
Example:
|x| < 2
x-2 and x2
Greater than Absolute Value Inequalities
If the absolute value inequality contains a greater than symbol (>=, >)
Write two separate inequalities to solve.
1. Drop the absolute value bars.
2. Create a second inequality with the flipped inequality symbol, opposite value, and the word “and” in between.
The graph has two parts in opposite directions.
Example:
|x| >= 6
X<=-6 or x >=6