Inequalities That Have No Solution

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the wild world of mathematics, specifically tackling those tricky inequalities that sometimes just don't play nice. You know, the ones that leave you scratching your head, wondering if you made a mistake. Well, we're here to clear that up and show you exactly which inequality from our lineup has absolutely no solution. So, grab your calculators, your favorite thinking snacks, and let's break down these problems step-by-step. We'll make sure you understand why an inequality might have no solution and how to spot it from a mile away. Get ready to conquer these mathematical beasts!

Understanding Inequalities and Solutions

Alright, so before we jump into solving, let's quickly chat about what inequalities even are and what it means for them to have a solution. Think of an inequality like a comparison between two expressions. Instead of saying two things are equal (like in an equation), we're saying one is greater than, less than, greater than or equal to, or less than or equal to the other. Our main goal when solving an inequality is to find the values of the variable (usually 'x') that make the statement true. For example, in the inequality $x > 5$, any number bigger than 5 is a solution. You could plug in 6, 7, 100, and the statement would hold true. Now, sometimes, when you go through the process of solving an inequality, you end up with a statement that is always false, no matter what number you plug in for 'x'. This is what we call an inequality with no solution. It's like trying to prove that 1 is equal to 2; it's just not possible! We'll be looking for one of these impossible scenarios in the options provided. The key is to simplify each inequality down to its most basic form and see what kind of statement we're left with. Are we left with a true statement that works for all 'x' (infinite solutions)? Or are we left with a false statement that works for no 'x' (no solution)? Let's get to the nitty-gritty!

Solving Inequality A: $6(x+2) > x-3$

Okay, team, let's kick things off with inequality A: $6(x+2) > x-3$. Our mission here is to isolate 'x' and see what kind of condition we end up with. First things first, we need to distribute that 6 on the left side. So, $6 imes x$ is $6x$, and $6 imes 2$ is $12$. Our inequality now looks like this: $6x + 12 > x - 3$. Now, we want all the 'x' terms on one side and the constants on the other. Let's subtract 'x' from both sides to move the 'x' term from the right to the left. $6x - x$ gives us $5x$. So, we have $5x + 12 > -3$. Next, let's move the constant term, 12, to the right side by subtracting 12 from both sides. $5x > -3 - 12$. This simplifies to $5x > -15$. The final step is to get 'x' all by itself by dividing both sides by 5. Since 5 is a positive number, the inequality sign stays the same. $x > -15 / 5$. This gives us $x > -3$. This is a perfectly valid solution! It means any number greater than -3 will make the original inequality true. For example, if we plug in $x=0$, we get $6(0+2) > 0-3$, which is $12 > -3$. That's true! So, inequality A does have a solution. We keep hunting!

Solving Inequality B: $3 + 4x

less 2(1 + 2x)$

Now, let's tackle inequality B, which is written as $3 + 4x less 2(1 + 2x)$. The symbol '$less$' means 'less than or equal to'. So, we can rewrite this as $3 + 4x less 2(1 + 2x)$. Just like before, we need to simplify. Let's start by distributing the 2 on the right side: $2 imes 1$ is $2$, and $2 imes 2x$ is $4x$. So, the inequality becomes $3 + 4x less 2 + 4x$. Now, let's try to get our 'x' terms together. If we subtract $4x$ from both sides, we get: $3 + 4x - 4x less 2 + 4x - 4x$. This simplifies down to $3 less 2$. Wait a minute! Is 3 less than or equal to 2? Absolutely not! This statement, $3 less 2$, is false. And because the variable 'x' completely disappeared during the simplification process, and we were left with a statement that is always false, this means that no value of x can ever make the original inequality true. This, my friends, is our inequality with no solution! We found it! But just to be absolutely sure, let's quickly check the other options to solidify our understanding. It's always good practice to double-check, especially in math!

Solving Inequality C: $-2(x+6)

less -10$

Let's examine inequality C: $-2(x+6) less -10$. This uses the 'less than or equal to' symbol again. First step, distribute the -2 on the left side. $-2 imes x$ is $-2x$, and $-2 imes 6$ is $-12$. So, we have $-2x - 12 less -10$. Now, we want to get the 'x' term by itself. Let's add 12 to both sides: $-2x - 12 + 12 less -10 + 12$. This simplifies to $-2x less 2$. Now, here's a crucial part: we need to divide by -2 to isolate 'x'. Remember the rule, guys? When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. So, $-2x / -2$ becomes $x$, and the $less$ sign flips to $>$. And $2 / -2$ is $-1$. So, our simplified inequality is $x > -1$. This means that any number greater than -1 is a solution. For instance, if $x=0$, we get $-2(0+6) less -10$, which is $-12 less -10$. This is true! Therefore, inequality C does have a solution. Phew, glad we checked!

Solving Inequality D: $x - 9 < 3(x - 3)$

Finally, let's analyze inequality D: $x - 9 < 3(x - 3)$. This is the last one on our list, and we want to see if it also has no solution, or if it does. Let's start by distributing the 3 on the right side: $3 imes x$ is $3x$, and $3 imes -3$ is $-9$. So, the inequality becomes $x - 9 < 3x - 9$. Now, let's get all the 'x' terms on one side. Subtract 'x' from both sides: $x - x - 9 < 3x - x - 9$. This gives us $-9 < 2x - 9$. Next, let's move the constant term, -9, to the left side by adding 9 to both sides: $-9 + 9 < 2x - 9 + 9$. This simplifies to $0 < 2x$. To isolate 'x', we divide both sides by 2. Since 2 is positive, the inequality sign stays the same: $0 / 2 < 2x / 2$. This results in $0 < x$, or more commonly written as $x > 0$. This inequality has solutions! Any number greater than 0 makes the original statement true. Let's test $x=1$: $1 - 9 < 3(1 - 3)$, which is $-8 < 3(-2)$, so $-8 < -6$. This is a true statement. So, inequality D does have a solution. Excellent!

The Verdict: Which Inequality Has No Solution?

After meticulously working through each inequality, we've reached our conclusion. We examined inequality A ($6(x+2) > x-3$), which simplified to $x > -3$, meaning it has infinitely many solutions. We then tackled inequality B ($3+4x less 2(1+2x)$), and after simplifying, we were left with the statement $3 less 2$. Since this is a false statement and the variable 'x' disappeared, we identified this as the inequality with no solution. We continued our analysis with inequality C ($-2(x+6) less -10$), which simplified to $x > -1$, confirming it has solutions. Lastly, we looked at inequality D ($x-9 < 3(x-3)$), which simplified to $x > 0$, also indicating it has solutions. Therefore, the inequality that has no solution is B. $3+4x less 2(1+2x)$. It’s all about simplifying and recognizing when you’re left with a universally false statement. Keep practicing these, guys, and you'll be spotting them like a pro in no time! Thanks for joining us on Plastik Magazine for this math adventure!