I'm stuck on this question, and really need help.
Question: True or false? Explain your answer.
The inequality −2(x + 10) ≥ 75 says the same thing as −2x − 20 ≥ 75. I can multiply by -2 on the left side without reversing the inequality symbol.
Answer: True. You are distributing the -2 in the parenthesis.
If anyone has anything to add to it, please do (I'm not sure if this is the complete answer.)
I agree
False. Multiplying by negative values reverses the direction. It is really just a shorthand way of swapping sides.
-2x-20 >= 75
add 2x to get
-20 >= 75+2x
subtract 75
-95 >= 2x
Now the 2x is on the small side of the inequality, so it's the same as
2x <= -95
Oops. I read ahead. You (and of course, Reiny!) are correct
Thank you guys so much, I appreciate the help!
You are correct in saying that the inequality −2(x + 10) ≥ 75 can be simplified to −2x − 20 ≥ 75.
To explain why this is true, let's follow the steps to simplify the inequality.
Step 1: Distribute the -2 to the terms inside the parentheses.
When you multiply -2 by x, you get -2x, and when you multiply -2 by 10, you get -20. So the inequality becomes -2x - 20 ≥ 75.
Step 2: Compare the simplified inequality to the original inequality.
After simplifying, we can see that the two inequalities have the same structure, with the same terms on both sides of the inequality symbol.
Step 3: Examine the effect of multiplying both sides of the inequality by -2.
In this case, we are multiplying the entire inequality by -2, which is a negative number. When you multiply or divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality symbol.
However, in our case, we multiplied only one side of the inequality by -2, and not both sides. Since we didn't reverse the direction of the inequality symbol, it means that the statement -2(x + 10) ≥ 75 says the same thing as -2x - 20 ≥ 75.
Therefore, the statement "True or false? The inequality −2(x + 10) ≥ 75 says the same thing as −2x − 20 ≥ 75, and you can multiply by -2 on the left side without reversing the inequality symbol" is true.