Solve For W: $\frac{w}{5}-9 \geq 16$

Hey guys! Today we're diving into a super common math problem that pops up all the time in algebra: solving inequalities. Specifically, we're going to tackle this one: $\frac{w}{5}-9 \geq 16$. Don't let the fraction or the "greater than or equal to" symbol scare you off; it's actually pretty straightforward once you break it down. We'll walk through it step-by-step, making sure you understand why we do each thing. The goal is to isolate the variable, $w$, on one side of the inequality, just like you would with an equation, but with a little twist to keep in mind for when you multiply or divide by negative numbers. So, grab your notebooks, maybe a snack, and let's get this $w$ figured out!

Understanding Inequalities

Alright, let's kick things off by talking about what an inequality actually is. Unlike an equation, which states that two things are exactly equal (like $x = 5$), an inequality shows that one side is greater than, less than, greater than or equal to, or less than or equal to the other side. Think of it like a comparison. For example, if you have 5 apples and your friend has 3, you have more apples than your friend. In math terms, we'd say $5 > 3$. The inequality signs we use are: $>$ (greater than), $<$ (less than), $\geq$ (greater than or equal to), and $\leq$ (less than or equal to). Solving an inequality means finding all the possible values of the variable (in our case, $w$) that make the statement true. So, for $\frac{w}{5}-9 \geq 16$, we're looking for all the numbers $w$ that, when you plug them into the left side, result in a number that is 16 or bigger. It's like finding a whole range of solutions, not just a single number. This is a key difference from solving basic equations. The process is very similar to solving equations, where we use inverse operations to isolate the variable. However, there's one crucial rule to remember: if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. We'll keep that in mind as we go through our problem. For now, let's focus on the steps to get $w$ by itself in $\frac{w}{5}-9 \geq 16$.

Step 1: Isolate the Term with 'w'

Okay, team, the first major goal when we're solving any inequality (or equation, for that matter) is to get the term containing our variable as isolated as possible. In our problem, $\frac{w}{5}-9 \geq 16$, the term with $w$ is $\frac{w}{5}$. Right now, it's got a '-9' hanging out with it. To get rid of that '-9', we need to do the opposite operation, which is adding 9. And just like in a regular equation, whatever we do to one side of the inequality, we must do to the other side to keep the balance. So, we're going to add 9 to both the left side and the right side of our inequality.

Here's how it looks:

$\frac{w}{5} - 9 + 9 \geq 16 + 9$

When we simplify this, the '-9' and '+9' on the left side cancel each other out (because $-9 + 9 = 0$), leaving us with just $\frac{w}{5}$. On the right side, $16 + 9$ equals $25$. So, our inequality now looks much simpler:

$\frac{w}{5} \geq 25$

See? We've successfully moved the constant term (-9) away from the variable term. This is a huge step! You've used the addition property of inequality, which is basically saying you can add or subtract the same number from both sides without changing the inequality's truth. This property is super handy for simplifying the problem and getting us closer to finding the value of $w$. Remember, every step we take is about undoing operations to isolate $w$. We started with a two-step process (division by 5 and subtraction of 9), and by tackling the subtraction first, we've reduced it to a single-step problem involving division.

Step 2: Isolate 'w'

Now that we've got $\frac{w}{5} \geq 25$, our next mission is to get $w$ completely by itself. Right now, $w$ is being divided by 5. To undo division, we use the inverse operation: multiplication. So, we need to multiply both sides of the inequality by 5. This is where we need to be extra careful. We are multiplying by a positive number (5), so we do not need to flip the inequality sign. If we were multiplying or dividing by a negative number, that's when the flip happens, but not today!

Let's multiply both sides by 5:

$5 \times \frac{w}{5} \geq 5 \times 25$

On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out ($5/5 = 1$), leaving us with just $w$. On the right side, we need to calculate $5 \times 25$. That gives us $125$.

So, our final inequality becomes:

$w \geq 125$

And there you have it! We've successfully solved the inequality. This result, $w \geq 125$, means that any number that is 125 or greater will satisfy the original inequality $\frac{w}{5}-9 \geq 16$. For example, if $w=125$, then $\frac{125}{5}-9 = 25-9 = 16$, which is indeed $\geq 16$. If we try $w=130$, we get $\frac{130}{5}-9 = 26-9 = 17$, which is also $\geq 16$. Pretty cool, right? You've used the multiplication property of inequality here, which, along with the addition property, allows you to manipulate inequalities while maintaining their validity. Always remember that crucial rule about flipping the sign if you multiply or divide by a negative – it's the most common pitfall, but once you've got it down, solving inequalities becomes a breeze. Keep practicing, and you'll be an inequality pro in no time!

Verifying the Solution

It's always a good idea, especially when you're starting out, to verify your solution to make sure you didn't make any sneaky errors. We found that $w \geq 125$. To check this, let's pick a few values for $w$ and plug them back into the original inequality: $\frac{w}{5}-9 \geq 16$.

1. Test a value that should work (a value $\geq 125$):

Let's try $w = 130$. (We already did this briefly, but let's do it formally).

$\frac{130}{5} - 9 \geq 16$

$26 - 9 \geq 16$

$17 \geq 16$

This is true. Awesome! Our solution seems to be holding up.

2. Test the boundary value ($w = 125$):

This is the