why does the inequality sign change when both sides are multiplied or dived by a negative number?
Does this happen with equations? why or why not?
write an inequality & solve it use both the multiplication & addition properties of inequalities.
Thnk you. I have no clue!
When both sides of an inequality are multiplied or divided by a negative number, the inequality sign changes direction. This is because multiplying or dividing by a negative number essentially flips the order of the numbers on the number line.
For example, consider the inequality 4 > 2. If we multiply both sides by -1, we get -4 < -2, where the inequality sign changes from ">" to "<".
On the other hand, when solving equations, the inequality sign does not change direction when both sides are multiplied or divided by a negative number. This is because in equations, we are trying to find the exact value of the variable, rather than comparing two values.
Now, let's write and solve an inequality using both the multiplication and addition properties:
Let's say we have the inequality: 3x + 5 > 8
1. To isolate the variable term, we subtract 5 from both sides:
3x > 8 - 5
3x > 3
2. Next, we divide both sides by 3 to get the value of x:
x > 3/3
x > 1
So, the solution to the inequality is x > 1.
The inequality sign changes when both sides are multiplied or divided by a negative number because it affects the direction of the inequality.
To understand why this happens, let's consider the inequality sign ">". This sign represents a relationship where the value on the left is greater than the value on the right.
When you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. For example, if we have "a > b" and we multiply or divide both sides by a positive number, we still have "a > b". This is because multiplying or dividing by a positive number does not change the order of the numbers.
However, when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes. For example, if we have "a > b" and we multiply or divide both sides by a negative number, the inequality becomes "a < b". This is because multiplying or dividing by a negative number reverses the order of the numbers.
This property only applies to inequalities, not equations. In equations, the goal is to find the value(s) of the variable(s) that make both sides equal. The equality sign "=" indicates that both sides of the equation are exactly the same. In equations, the operations performed on both sides must be done in a way that maintains equality, such as adding or subtracting the same number from both sides, or multiplying or dividing both sides by the same nonzero number.
So, let's write and solve an inequality using both the multiplication and addition properties:
We'll use the inequality "2x + 5 > 11" and solve it step by step.
1. Start with the original inequality: 2x + 5 > 11
2. Subtract 5 from both sides to isolate the term with x: 2x + 5 - 5 > 11 - 5
Simplifying gives: 2x > 6
3. Now, let's divide both sides by 2 to solve for x:
(2x)/2 > 6/2
Simplifying gives: x > 3
Therefore, the solution to the inequality "2x + 5 > 11" using both the multiplication and addition properties of inequalities is x > 3.
I hope this explanation helps! Let me know if you have any further questions.