Why does the inequality sign change when both sides are multiplied or divided by a negative number? Does this happen with equations? Why or why not? Write an inequality for your classmates to solve. In your inequality, use both the multiplication and addition properties of inequalities.

by multiplying/dividing by a negative number, you are distributing that negative to every part of the equation, making everything effected by the negative opposite. In order to make an inequality sign opposite, you must flip it. with equations you still must multiply/divide everything by the negative, but you can not change an = they are not affected.

-3x+18>21

Ah, the magic of inequality signs! Well, when you multiply or divide both sides of an inequality by a negative number, the inequality sign changes its direction. Why? It's almost like the negative number is whispering in the inequality sign's ear, "Hey, buddy, let's flip things around!" And the inequality sign obediently follows suit, just to keep things balanced.

But when it comes to equations, it's a different story. Equations are all about finding the value that makes both sides equal. When you multiply or divide both sides by a negative number, the equation sign doesn't change because you're still aiming for equality, not inequality. It's like saying, "Hey, we're all in this together, let's keep striving for unity!"

Now, let me give you an inequality to solve:

3x + 5 > 2x + 10

Go ahead, put on your problem-solving hat and find that x value that makes this wacky inequality true! Good luck, my friends!

The inequality sign changes when both sides are multiplied or divided by a negative number because multiplying or dividing by a negative number changes the direction of the inequality. This happens because the negative sign flips the sign of the inequality.

However, this does not happen with equations. In equations, the goal is to find the value that makes both sides equal. Multiplying or dividing both sides of an equation by a negative number would change the values and make the equation false.

Here's an inequality for you to solve using both the multiplication and addition properties of inequalities:

3x + 2 > 8

To solve this inequality, you can follow these steps:
1. Start by subtracting 2 from both sides of the inequality:
3x + 2 - 2 > 8 - 2
3x > 6

2. Next, divide both sides of the inequality by 3:
(3x)/3 > 6/3
x > 2

So, the solution to the inequality is x > 2.

The inequality sign changes when both sides of the inequality are multiplied or divided by a negative number because multiplying or dividing by a negative number changes the direction of the inequality.

Let's start with an example to better understand why this happens. Suppose we have the inequality 2 < 3. If we multiply both sides by -1, we get -2 > -3. Notice that the inequality sign changes from < to >. This is because multiplying by -1 flips the order of the numbers on a number line. In this case, both 2 and 3 are positive numbers, and -2 and -3 are negative numbers, so their positions on the number line are reversed, and therefore the inequality sign changes.

The same principle applies to dividing by a negative number. If we have the inequality 4 < 8 and divide both sides by -2, we get -2 > -4. Again, the inequality sign changes because dividing by a negative number flips the order of the numbers on a number line.

However, when it comes to equations, the situation is different. When we multiply or divide both sides of an equation by a negative number, the inequality sign does NOT change. This is because when we solve an equation, we are looking for a number that makes both sides of the equation equal. Multiplying or dividing by a negative number does not change the equality of the equation. For example, if we have the equation 2x = 6 and divide both sides by -2, we get x = -3. The inequality sign does not change because we are not comparing two quantities, but finding a solution that satisfies the equation.

Now, let's create an inequality for your classmates to solve using both the multiplication and addition properties of inequalities:

5(x - 3) + 2 > 3x - 4

To solve this inequality, we will first apply the multiplication property to distribute the 5:

5x - 15 + 2 > 3x - 4

Next, we will combine like terms by adding 15 and 4 to both sides:

5x + 2 > 3x + 11

Finally, we will apply the subtraction property by subtracting 3x from both sides:

2x + 2 > 11

Now, your classmates can solve for x by subtracting 2 from both sides:

2x > 9

And finally, dividing both sides by 2:

x > 4.5

Therefore, the solution to the inequality is x > 4.5.

x<-1