Unlock Equivalent Inequalities: Find Your Match!

Hey math whizzes! Ever feel like you're staring at a bunch of inequalities and wondering which ones are secretly telling the same story? Today, we're diving deep into the world of equivalent inequalities, specifically focusing on a cool target: $r > -11$. We'll break down each option, put 'em to the test, and figure out which ones are the true twins of our original inequality. Get ready to level up your inequality game, guys!

Understanding Equivalent Inequalities

First off, what does it really mean for inequalities to be equivalent? Think of it like this: two inequalities are equivalent if they have the exact same solution set. No matter what numbers you plug in, if a number makes one inequality true, it'll make the other one true too, and vice versa. It's like having two different-looking keys that both unlock the same door. In our case, we're looking for inequalities that are true for all values of 'r' that are greater than -11. So, if $r$ is 10, it's greater than -11. If $r$ is -5, it's greater than -11. If $r$ is -10.9, it's greater than -11. We need our equivalent inequalities to hold true for all these numbers and exclude anything less than or equal to -11.

There are a few key moves we can make to transform an inequality without changing its solution set. These are our golden rules:

1. Adding or Subtracting the Same Number: You can add or subtract any number from both sides of an inequality, and it remains equivalent. For example, $x > 5$ is equivalent to $x + 2 > 7$ or $x - 1 > 4$.
2. Multiplying or Dividing by the Same Positive Number: Multiplying or dividing both sides by a positive number keeps the inequality sign the same. So, $x > 5$ is equivalent to $2x > 10$ or $x/3 > 5/3$.
3. Multiplying or Dividing by the Same Negative Number: This is the tricky one, guys! When you multiply or divide both sides by a negative number, you must flip the inequality sign. For instance, $x > 5$ is equivalent to $-2x < -10$ (we divided by -2 and flipped the $>$ to $<$) or $x/(-1) < 5/(-1)$, which simplifies to $-x < -5$. This rule is super important and often trips people up, so keep it front and center in your math brain!

Our mission is to take each of the given options and see if we can transform it, using these rules, into our target inequality, $r > -11$. Alternatively, we can take $r > -11$ and transform it into one of the options. Let's get started!

Evaluating Each Inequality Option

Alright team, let's put on our detective hats and examine each potential candidate for equivalence to $r > -11$. We'll go through them one by one, applying our trusty inequality rules.

Option 1: $-r < 11$

Our first contender is $-r < 11$. We want to isolate 'r' here. What's the quickest way to get 'r' by itself? We need to get rid of that negative sign in front of 'r'. Remember our rule about multiplying or dividing by a negative number? That's exactly what we need to do here! We're going to multiply both sides of the inequality by -1. And what happens when we do that? You got it – we flip the inequality sign!

So, starting with $-r < 11$, if we multiply both sides by -1, we get:

$(-1) imes (-r) \_? \_ (-1) imes 11$

This simplifies to:

$r \_? \_ -11$

Since we multiplied by a negative number, we must flip the inequality sign from $<$ to $>$. So, the equivalent inequality is:

$r > -11$

Boom! This inequality, $-r < 11$, is exactly equivalent to our target inequality $r > -11$. So, mark this one down, folks!

Option 2: $3r < -33$

Next up, we have $3r < -33$. Our goal is to get 'r' by itself. Right now, 'r' is being multiplied by 3. To undo multiplication, we use division. We'll divide both sides of the inequality by 3. Since 3 is a positive number, we don't need to worry about flipping the inequality sign. It stays the same.

Starting with $3r < -33$, let's divide both sides by 3:

$(3r) / 3 \_? \_ -33 / 3$

This simplifies to:

$r \_? \_ -11$

Since we divided by a positive number (3), the inequality sign remains the same: $<$. So, the resulting inequality is:

$r < -11$

Now, let's compare this to our target inequality, $r > -11$. Are they the same? Nope! $r < -11$ means 'r' is any number less than -11 (like -12, -100, -1000), while $r > -11$ means 'r' is any number greater than -11 (like -10, 0, 100). These are completely opposite solution sets. Therefore, $3r < -33$ is NOT equivalent to $r > -11$.

Option 3: $3r > -33$

Let's tackle $3r > -33$. Again, our mission is to isolate 'r'. We'll divide both sides by 3. Since 3 is positive, the inequality sign stays the same.

Starting with $3r > -33$, divide both sides by 3:

$(3r) / 3 \_? \_ -33 / 3$

This simplifies to:

$r \_? \_ -11$

Because we divided by a positive number (3), the inequality sign remains $>$. So, the resulting inequality is:

$r > -11$

Bingo! This inequality, $3r > -33$, simplifies down to $r > -11$. That means it has the exact same solution set as our target inequality. So, this one is also a winner, guys!

Option 4: $-3r < 33$

Moving on to $-3r < 33$. Our goal is to get 'r' alone. 'r' is currently multiplied by -3. To isolate 'r', we need to divide both sides by -3. Now, here comes that crucial rule again: when you divide by a negative number, you must flip the inequality sign.

Starting with $-3r < 33$, divide both sides by -3:

$(-3r) / (-3) \_? \_ 33 / (-3)$

This simplifies to:

$r \_? \_ -11$

Since we divided by a negative number (-3), we need to flip the inequality sign from $<$ to $>$. So, the equivalent inequality is:

$r > -11$

Yes! This inequality, $-3r < 33$, also simplifies perfectly to $r > -11$. It has the same solution set. So, this is another equivalent inequality we need to check off our list!

Option 5: $-3r > 33$

Our final candidate is $-3r > 33$. To isolate 'r', we will divide both sides by -3. And remember the golden rule: dividing by a negative number means we have to flip the inequality sign.

Starting with $-3r > 33$, divide both sides by -3:

$(-3r) / (-3) \_? \_ 33 / (-3)$

This simplifies to:

$r \_? \_ -11$

Since we divided by a negative number (-3), we flip the inequality sign from $>$ to $<$. So, the resulting inequality is:

$r < -11$

Let's compare this to our target $r > -11$. Are they the same? Absolutely not! $r < -11$ represents all numbers less than -11, whereas $r > -11$ represents all numbers greater than -11. They are opposites. Therefore, $-3r > 33$ is NOT equivalent to $r > -11$.

The Grand Finale: Which Ones Are Equivalent?

Alright, math adventurers, we've put all the options under the microscope, applied our inequality superpowers, and deciphered their true meanings. Let's recap our findings:

• $-r < 11$ simplified to $r > -11$. Equivalent!
• $3r < -33$ simplified to $r < -11$. Not Equivalent!
• $3r > -33$ simplified to $r > -11$. Equivalent!
• $-3r < 33$ simplified to $r > -11$. Equivalent!
• $-3r > 33$ simplified to $r < -11$. Not Equivalent!

So, to answer the big question: "Which of these inequalities are equivalent to $r > -11$?", the ones you should check are:

• □ $-r<11$
• □ $3r>-33$
• □ $-3r<33$

Great job sticking with it, guys! Mastering equivalent inequalities is a fundamental skill that will serve you well in all sorts of mathematical challenges. Keep practicing, keep questioning, and you'll be an inequality ninja in no time! See you in the next math adventure!