Truth Values of and/or
Solution sets of equations and inequalities
In this lesson we will learn about Compound Inequalities and what it means when we have two or more inequalities with “and/or” statements. We will learn there is a slight difference between the truth of a statement with “and/or” in our spoken language and the truth of a statement with “and/or” in the mathematical language.
Below we will look at the truth of “and/or” statements in our spoken language and mathematical language.
An “and” statement in our spoken language
If I said “I am from Oregon and my name is Julian”, for this statement to be true, both parts of the statement would have to be true. If I said “I am from the North Pole and my name is Julian”, even though my name might be Julian, if I am not from the North Pole, then the whole statement is false. So, for “and” statements both parts of the statement must be true for the whole statement to be true.
An “and” statement in Mathematical Language
It follows the same rules as in our spoken language. So, if I said “2 > 1 and 100 > 10” this is a true statement if both parts of the statement are true. We know 2 is bigger than 1 and 100 is bigger than 10, so as both parts of the statement are true, the whole statement is true.
An “or” statement in our spoken language:
If I said “My name is Julian or I am from Oregon”, and both parts of that statement were true, then in our spoken language we would think that statement is false, as it is like saying “either I am this or I am that”. “Or” statements are where we have a difference between spoken language and mathematical language.
An “or” statement in Mathematical Language:
“Or” statements in math follow a different rule than in our spoken language. In Mathematical language, an “or” statement is true as long as one part of the statement is true. It is also true if both parts of the statement are true. If we said “3 > 1 or 99 > 9 ”, it doesn't matter that both parts are true, at least one part of the “or” statement is true, therefore it is mathematically true.
The solutions in the above animation are repeated with a more detailed explanation below.
Question 1
Look at the following compound inequality statements and determine if they are mathematically true.
4 < 5 and 7 > -1
For these questions, first, find out what you know about each part of the statement. Figure out if each part of the statement is true or false, then look at whether it’s an “and” or an “or” statement and then determine if the whole statement is true.
We know that:
4 is less than 5
7 is greater than -1
As it is an “and” statement, both parts need to be true for the whole statement to be true
Both parts of the statement are true, so the statement is true.
We know that:
Don't be thrown off by the ≤ sign, we are still looking at if each part of the statement is true, so we check if -6 is less than or equal to -3
-6 is less than or equal to -3
5 is not greater than 104
This is an “or” statement, so only one part needs to be true. Only one part is true, so, the statement is true.
We know that:
5 is not less than -2
6 is greater than / equal to 1
As this is an “and” statement, both parts need to be true. One part is true and one part is false, therefore, the statement is false.
We know that:
-2 is greater than -7
16 is greater than 12
As this is an “or” statement, only one part needs to be true, however as we remember for “or” statements, both parts can be true as well. We see that both parts are true, therefore, this statement is true.
We know that:
7 is not less than 8
10 is not less than / equal to 11
As this is an “or” statement, only one part needs to be true. Neither part of this statement is true, therefore, this statement is false.
Now we will move onto checking the truth of inequalities that have algebraic expressions with certain values of variables. Let's remind ourselves of the definition for solutions and solution sets.
The solution / solution set of an inequality or expression is true if it makes the inequality or expression true. It is not a solution / solution set if it makes the inequality or equation false.
So, we will be checking or solving for variables in inequalities to see if they are true. If they are true, they will be part of the solution set. We will be checking inequalities that involve “and/or” statements by applying the methods we learned above.
The solutions in the above animation are repeated with a more detailed explanation below.
Question 2
Write in words the solution set to the compound inequalities (this is to give an idea of what the solution set is), then plot the solution set of the compound inequality on a number line.
-1 ≤ x and x < 5
First, we write in words the solution set.
So, we are looking for all the values of x that make the statement true.
We have an “and” statement, so our solution set will be the values that satisfy both parts of the statement. We need all the values that satisfy -1 ≤ x and x<5,
This means the solution set is all the values that are greater than or equal to -1 and less than 5. This will help us plot the solution set on a number line.
Start by writing two dots above our restricted values, -1 and 5.
As we remember from the previous lesson, when writing solution sets on number lines, if the restricted value is a “greater/less than or equal to” (≥ / ≤) value, we write a solid dot. If it is just a “greater/less than” (< / >) value, we write a hollow dot. So above -1 we write a solid dot and above 5 we write a hollow dot.
Now, to finish our number line, we write a line over top of all the values that are in the solution, connecting our two dots.
These are all the numbers that are greater than or equal to -1 and less than 5, so we write:
There we have our solution set written on a number line. This gives us a visual of all the values that make our statement -1 ≤ x and x < 5 true.
Let's repeat this with another example
Roughly write what the solution set is in words.
So, we are looking for all the values that are greater than or equal to 3 or less than 0. This means our solution set is all the values that are greater than or equal to 3 or less than 0.
For our number line, start by writing a solid dot above 3, and a hollow dot above 0.
Now, as we have two parts to this solution set (all the values greater than or equal to 3 or all the values less than 0), we will have two lines.
Also, both parts of this solution set have infinite values.
There is an infinite amount of numbers greater than or equal to 3.
There is an infinite amount of numbers less than 0.
So, both lines drawn on the number line will have arrows at the end to represent that the number line is infinite.
So, let us write our number line over the values that are in our solution set with an arrow at the end of each line starting at each of our dots drawn.