We are learning inequalities and I just did some homework but the teacher returned it saying I did some of them wrong.
1. -2x - 10 < 2
2. -2x + 15 < 17
I didn't flip the inequality sign when I divided by (-2).
But why do we always have to flip the sign when we multiply or divide by a negative number?
In google type:
inequalities and negative numbers
You will find couple sites.
mathforum is good.
1.
-2 x - 10 < 2
-2 x < 2 + 10
-2 x < 12 Divide both sides with -2
x > 12 / - 2
x > - 6
2.
-2x + 15 < 17
-2 x < 17 - 15
-2 x < 2 Divide both sides with -2
x > 2 / -2
x > -1
When solving inequalities, it is important to understand why we need to flip the inequality sign when multiplying or dividing by a negative number.
To explain this, let's start with the first inequality:
-2x - 10 < 2
To solve for x, we would first subtract 10 from both sides:
-2x < 2 + 10
-2x < 12
Now, in order to isolate x, we need to divide both sides of the equation by -2:
(-2x) / -2 > 12 / -2
x > -6
This is where the flipping of the inequality sign comes into play. Since we divided both sides by a negative number (-2), we need to flip the sign to maintain the inequality.
If we didn't flip the inequality sign and simply wrote x < -6, it would be incorrect.
Now, let's consider the second inequality:
-2x + 15 < 17
Following the same steps, we subtract 15 from both sides:
-2x < 17 - 15
-2x < 2
Then, dividing by -2:
(-2x) / -2 > 2 / -2
x > -1
Again, since we divided by a negative number (-2), we must flip the inequality sign. The correct solution is x > -1.
So, to summarize, when multiplying or dividing both sides of an inequality by a negative number, we need to flip the inequality sign to maintain the correct direction of the inequality. This ensures that our solution represents all the values that satisfy the original inequality.