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roger
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hello
please could someone explain to me inequalities ?
I don't understand how it works
Roger
please could someone explain to me inequalities ?
I don't understand how it works
Roger
arildno said:All right then:
ASSUMING tthe inequality is TRUE, the inequality you gain by adding the SAME number "y" to each side, must ALSO be TRUE (agreed?)
We choose therefore (because it is SIMPLIFYING) to add the number y=3x to both sides of the inequality.
Our new inequality, which must have the same truth value as our original inequality, is therfore:
[tex]6x+4+3x\geq{-3x}+9+3x[/tex]
Or, simplifying both sides:
[tex]9x+4\geq9[/tex]
Now, do you have any ideas as to how to simplify even further?
arildno said:What I meant was:
Look at the following inequalities:
[tex]6x+4\geq9-3x (1)[/tex]
[tex]6x+4+3x\geq9-3x+3x (2)[/tex]
a) These inequalities are logically EQUIVALENT because:
ai)if (1) is true then (2) is true
AND
aii)if (2) is true then (1) is true.
(Clearly, it is also correct that if (1) is FALSE, then (2) is false as well, and vice versa).
To rephrase this in terms of "x":
The set of x-values which makes (1) TRUE is the same set which makes (2) TRUE, and the set of x-values making (1) FALSE is the same set which makes (2) FALSE.
b) HOWEVER:
You cannot say, for example that the left-hand-sides of (1) and (2) are EQUAL to each other, right?
([tex]6x+4\neq6x+4+3x[/tex])
Nor can you say that the right-hand sides of (1) and (2) are EQUAL to each other.
([tex]9-3x\neq9-3x+3x[/tex])
c) Hence, you are entitled to say that by adding 3x to both sides of (1), you GAIN a new, but equivalent inequality (that is, (2)).
To take a simple example:
Consider the following TRUE inequality:
[tex]2\leq7 (3)[/tex]
Adding 2 to to both sides, yields the also TRUE inequality:
[tex]4\leq9 (4)[/tex]
That is, adding the same number on both sides PRESERVES the TRUTH VALUE of your original inequality, but do you consider (3) and (4) to be strictly the SAME inequality?
Since 2 isn't 4 and 7 isn't 9, I agree with you.roger said:I understand so far what you have written.
I think the last two inequalities are different ?
arildno said:Since 2 isn't 4 and 7 isn't 9, I agree with you.
But do you also agree with me on the issue that adding the same number to both sides of an inequality preserves the truth value of your original inequality?
That is, that the new and old inequalities are logically equivalent?
Inequalities are mathematical expressions that compare two quantities or values. They use symbols like <, >, ≤, or ≥ to show which value is greater or less than the other. For example, 3 < 5 is an inequality that shows that 3 is less than 5.
Inequalities and equations are both mathematical expressions, but they serve different purposes. Inequalities compare two values and show the relationship between them, while equations state that two values are equal. Inequalities have a range of possible solutions, while equations have only one solution.
Strict inequalities use symbols like < and >, and they indicate that the two values being compared are not equal. For example, 4 < 5 means that 4 is less than 5, but 4 is not equal to 5. Non-strict inequalities use symbols like ≤ and ≥, and they indicate that the two values being compared may be equal. For instance, 3 ≤ 3 means that 3 is less than or equal to 3.
To graph an inequality, you first need to determine the boundary line for the inequality. This line is usually a straight line, but it can also be a curved line in some cases. Then, you determine whether the solution is above or below the boundary line. If the solution is above the line, shade the area above the line. If the solution is below the line, shade the area below the line. The shaded area represents the solution set for the inequality.
Inequalities can be solved by using algebraic methods, such as addition, subtraction, multiplication, and division, to isolate the variable on one side of the inequality sign. However, it is important to note that the direction of the inequality sign must be switched if any of these operations are performed on both sides of the inequality. Additionally, inequalities can also be solved graphically by finding the intersection point of the boundary line and the solution set.