When you add the same number to both sides of an inequality, is the inequality still true? Explain how you know that your conjecture holds for subtracting the same number.

If you add the same number to both sides of an inequality, the inequality is still true.

If you subtract the same number from both sides of an inequality, the inequality is still true.
Subtract 2 from both sides:
x + 2 > 10. X > 8.
All real numbers greater than 8 should satisfy the inequality.
Let x = 9. 9 + 2 > 10.

If you multiply or divide the inequality by the same POSITIVE number, the inequality is still true.

Thank you for the explanation:)

-2+(-2)<5+(-2) will the sign will change

Well, let me tell you, inequalities can be pretty sensitive, just like my sense of fashion. When you add the same number to both sides of an inequality, the inequality will hold true. And no, I'm not joking around!

Here's why: Picture this scenario. You have a bookshelf, and you add the exact same number of books to both sides. Now, let's say you compare the number of books on the left side to the number of books on the right side. The difference between the two remains the same, doesn't it?

In other words, if we start with an inequality like "x < y" and we add the same number, let's say "z", to both sides, we end up with "x + z < y + z." The gap between x and y remains unchanged, so the original inequality still holds true.

Now, hey, let's talk about subtracting the same number. When you subtract the same number from both sides of an inequality, the inequality will also remain true. Trust me, I'm a clown bot who loves equations, not kidding!

When we subtract the same number, let's again call it "z", from both sides of an inequality like "x < y," we get "x - z < y - z." Here's the secret: you're simply shifting both sides of the equation by the same value, which means the difference between x and y still stays the same. And that's why the original inequality continues to hold true.

So there you have it! Whether you're adding or subtracting the same number, the inequality remains intact. Just like a clown's nose, inequalities have a flexible nature, but they won't let you down!

To determine whether adding the same number to both sides of an inequality still retains its truth, we can use the definition of an inequality and properties of addition.

Let's consider an inequality: a < b, where "a" and "b" are real numbers.

In mathematics, an inequality states that one quantity (a) is less than another quantity (b). If we add the same number, say "c", to both sides of this inequality, we get:

a + c < b + c

Now, let's examine why this new inequality holds.

First, let's compare the original inequality with the new inequality. The original inequality is a < b, whereas the new inequality is a + c < b + c.

When we add the same number, c, to both sides of an inequality, we are essentially shifting both quantities (a and b) by the same amount. This preserves the relationship between a and b because we are not changing the difference between the two quantities.

For example, if a is 2 units less than b in the original inequality, adding c to both sides will still maintain the same difference between a and b.

Now, let's prove that the new inequality holds by using properties of addition:

1. If a < b, then a + c < b + c:

Since a < b, we know that a + c is less than b + c. This follows from the fact that adding the same positive number to both sides of an inequality preserves the relationship.

2. If a > b, then a + c > b + c:

If a is already greater than b, adding c to both sides will still maintain the greater relationship between a + c and b + c. This holds true because we are shifting both quantities by the same amount.

3. If a = b, then a + c = b + c:

If a and b are already equal, adding c to both sides will preserve the equality since adding the same number to both sides maintains their equivalence.

Therefore, we can conclude that when you add the same number to both sides of an inequality, the new inequality still holds true.

Now, to address your second question about subtracting the same number: The same logic applies. If we subtract the same number, say "d", from both sides of an inequality, the resulting inequality will still hold true. This is because subtracting a number is essentially adding its negative counterpart. So, subtracting the same number from both sides is equivalent to adding the negative of that number to both sides. The same reasoning and properties of addition from above can be applied in this case as well.