Why does the inequality sign change when both sides are multiplied or divided by a negative number? Does this happen with equations? Explain
Multiplying by a negative number is just a quick way of moving things to the other side of the inequality. Consider
-2x < 100
Now, you could solve it without doing any negative multiplying at all: just add 2x to both sides to get
0 < 2x + 100
Now subtract the 100 to get
-100 < 2x
or,
2x > -100
changing the sense of the inequality just means that terms have switched sides in the expression as written.
thank you!
The inequality sign changes when both sides of an inequality are multiplied or divided by a negative number because it reflects the concept of inequality and preserving the order of numbers.
To understand why this happens, let's consider an example. Suppose we have the inequality 3 > -5. If we multiply both sides by -1, we get -3 < 5. The inequality sign changed from 'greater than' to 'less than' because multiplying both sides by -1 essentially reflects the number line.
When we multiply both sides of an inequality by a positive number, the order of the numbers remains the same. For example, if we have 2 > 1 and we multiply both sides by 3, we get 6 > 3. The inequality sign does not change because multiplication by a positive number preserves the order of the numbers.
However, when we multiply both sides of an inequality by a negative number, the order of the numbers is reversed. Let's take the inequality 2 > 1 again, but this time we multiply both sides by -3. We get -6 < -3. The inequality sign changes from 'greater than' to 'less than' because the negative multiplication flips the number line.
This concept does not apply to equations because equations require equality, not inequality. In equations, we are trying to find the value(s) that make both sides equal. Multiplying or dividing both sides of an equation by a negative number does not change the equality. For example, if we have 2x = 8, and we multiply both sides by -1, we get -2x = -8. The equation is still true and the equality is preserved.