Why does the inequality sign change when both sides are multiplied or divided by a negative number? Does this happen with equations? Explain why or why not? Write an inequality for your classmates to solve. In your inequality, use both the multiplication and addition properties of inequalities.
take any true inequality, say ...
4 < 10
now multiply both sides by 10
40 < 100 , still true
divide original by 2
2 < 5 , still true
multiply original by -10
-40 < -100 , FALSE!, how do we make it true ?
-40 > -100
divide original by -2
-2 < -5 , FALSE! , how do we make it true?
-2 > -5
what about equations?
5 = 5 , true
multiply by -1
-5 = -5, (did you notice how I switched the = sign from = to = ?? , lol)
Another way to think of this changing direction stuff is to leave the direction the same, and move the expressions. Suppose you have
-3x > 9
Rather than dividing by -3, let's get rid of the negative coefficient by adding and subtracting.
Add 3x to both sides:
-3x + 3x > 9 + 3x
0 > 9 + 3x
Now subtarct 9 from both sides
0 - 9 > 9 + 3x - 9
-9 > 3x
Now divide by +3 to get
-3 > x
So, what we have is
-3x > 9 changes to
-3 > x
and even though we haven't change direction of the actual symbol, we have done it logically by moving things to the other side of the arrow.
The inequality sign changes when both sides are multiplied or divided by a negative number because it reflects the concept of "flipping" the number line. When we multiply or divide both sides of an inequality by a negative number, it changes the direction of the inequality. This is because multiplying or dividing by a negative number effectively flips the sign of the number.
Let's consider an example to illustrate this:
Original inequality: x > 5
If we multiply both sides of the inequality by -1, we get:
-1(x) < -1(5) which simplifies to -x < -5
Now, let's talk about equations. When we multiply or divide both sides of an equation by a negative number, the sign does not change. This is because in an equation, we have an equal sign (=) which indicates that both sides are the same.
For example:
Original equation: 2x = 10
If we multiply both sides of the equation by -1, we get:
-1(2x) = -1(10) which simplifies to -2x = -10
Here, the sign does not change because both sides are equal and we are only working towards finding the value of x that satisfies the equation.
Now, to write an inequality for your classmates to solve, let's construct one:
Let's say we want to write an inequality using both the multiplication and addition properties.
Inequality: 3x + 2 > 8
Here, we have used the addition property by adding 2 to both sides of the inequality, and the multiplication property by multiplying both sides by 3. To solve this inequality, your classmates would need to simplify it and isolate the variable x.