Convert Inequality To Slope-Intercept Form

Hey guys! Ever stared at an inequality like $6x + 2y \leq -46$ and wondered, "What in the math world am I supposed to do with this?" Don't sweat it! Today, we're diving deep into how to wrangle these beasts and transform them into slope-intercept form. It's not as scary as it sounds, and once you get the hang of it, you'll be converting inequalities like a pro. We'll break it down step-by-step, making sure you understand every little bit. Think of slope-intercept form as your friendly guide to understanding the graph of your inequality. It tells you where to start (the y-intercept) and which way to go (the slope). So, grab your notebooks, get comfy, and let's make some math magic happen!

Understanding Slope-Intercept Form

So, what exactly is slope-intercept form? It's basically a way to write linear equations and inequalities that makes them super easy to understand and graph. For an equation, it looks like this: y = mx + b. Here, 'm' is your slope, which tells you how steep your line is and in which direction it's going (up or down, left or right). Think of it as the 'rise over run' of your line. The 'b' is your y-intercept, which is simply the point where your line crosses the y-axis. It's that crucial starting point. When we move to inequalities, like the one we're tackling, $6x + 2y \leq -46$, we'll be aiming for a similar structure, but instead of an equals sign, we'll have an inequality sign (like $\leq$, $\geq$, $<$, or $>$) and the 'y' will be isolated on one side. This isolated 'y' is key because it allows us to easily determine whether the area above or below the line needs to be shaded on our graph. The slope and y-intercept still give us the boundary line, and the inequality sign tells us which side of that line represents all the solutions to our problem. It's like getting a secret code to unlock the graphical representation of your inequality. Understanding this form is fundamental because it translates abstract mathematical expressions into visual, understandable graphs. Without it, plotting inequalities would be a guessing game, but with it, we have a clear roadmap. The slope, 'm', dictates the angle and direction of the line. A positive 'm' means the line goes up from left to right, while a negative 'm' means it goes down. A fractional slope, like 1/2, means for every 2 steps you move to the right, you move up 1 step. The y-intercept, 'b', is just as important; it's literally where the line intersects the y-axis. This point is $(0, b)$. Knowing these two pieces of information is like having the keys to the kingdom when it comes to graphing linear functions and inequalities. So, when we convert $6x + 2y \leq -46$ to slope-intercept form, our goal is to get it into the shape of $y \leq mx + b$ (or one of the other inequality variations). This will reveal the 'm' and 'b' values for the boundary line and tell us which side of the line contains the solutions.

Step-by-Step Conversion Process

Alright, team, let's get down to business with our inequality: $6x + 2y \leq -46$. Our mission, should we choose to accept it (and we totally should!), is to isolate the 'y' term. This is just like solving for 'y' in a regular equation, but we need to be extra careful with one thing: if we ever multiply or divide both sides by a negative number, we must flip the inequality sign. Keep that rule in your back pocket! First things first, we want to get the term with 'y' all by itself on one side. To do that, we need to move the '$6x$' term. Since it's currently positive '$6x$', we'll subtract '$6x$' from both sides of the inequality. This gives us: $2y \leq -46 - 6x$. Now, see how the '$6x$' is on the right side with the constant? That's okay for now. The next step is to get 'y' completely alone. Right now, 'y' is being multiplied by 2. To undo that multiplication, we need to divide both sides by 2. And here's where we pause and check: are we dividing by a positive or a negative number? In this case, it's positive 2, so hooray, we don't need to flip the inequality sign! So, dividing both sides by 2, we get: $y \leq \frac{-46 - 6x}{2}$. Now, we just need to simplify the right side. We can split the fraction: $y \leq \frac{-46}{2} - \frac{6x}{2}$. Performing the divisions, we arrive at our final slope-intercept form: $y \leq -23 - 3x$. But wait! Standard slope-intercept form is usually written with the '$x$' term first. So, let's just rearrange that slightly for the classic look: $y \leq -3x - 23$. And there you have it! We've successfully converted the original inequality into slope-intercept form. The slope '$m$' is -3, and the y-intercept '$b$' is -23. This tells us that our boundary line will have a slope of -3 and cross the y-axis at -23. Since the inequality sign is 'less than or equal to' ($\leq$), we know that all the points on the line and below the line are solutions to the original inequality. This transformation is crucial for visualizing and understanding the set of all possible solutions. It turns a complex expression into a clear graphical instruction. So, the process involves two main algebraic moves: getting the 'y' term isolated and then dividing to make the coefficient of 'y' equal to 1. Always, always keep an eye on that inequality sign, especially during division or multiplication.

Interpreting the Slope-Intercept Form

Now that we've got our inequality $6x + 2y \leq -46$ all tidied up into $y \leq -3x - 23$, what does this actually mean? This is the fun part where we unlock the visual story of the inequality! Remember, '$m$' is our slope and '$b$' is our y-intercept. In our case, the slope m = -3 and the y-intercept b = -23. The y-intercept, -23, tells us that the boundary line for this inequality crosses the y-axis at the point (0, -23). This is our starting point when we're thinking about graphing. The slope, -3, tells us about the steepness and direction of this boundary line. Since it's negative, the line will slant downwards as you move from left to right. Specifically, for every 1 unit you move to the right on the graph, the line goes down by 3 units. You can think of it as a