Solve Inequalities: A Visual Guide

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically tackling inequalities. You know, those cool mathematical statements that tell us when one thing is less than or greater than another? We're going to take a look at a specific problem, solve the inequality and graph the solution set, and then express our findings in two super important ways: set-builder notation and interval notation. We'll also make sure all our numbers are either nice, clean integers or simplified fractions. So, grab your notebooks, maybe a snack, and let's get this math party started!

Understanding the Inequality

Our main mission today is to solve the following inequality: $2-2(y-1)<-5 y+10$. Before we even start crunching numbers, let's break down what this means. We've got variables, constants, and these inequality signs. The goal is to find all the possible values of 'y' that make this statement true. Think of it like finding the magic numbers that will make the left side of the equation strictly less than the right side. It's not an equation that has just one answer; inequalities usually have a whole range of answers, which is why we need to graph them and use special notation to describe them. This inequality involves a variable 'y' on both sides, and some parentheses, which means we'll need to use our trusty distributive property to simplify things first. Don't sweat the small stuff, though; we'll go step-by-step, making sure we understand each part. The real beauty of inequalities is that they represent a set of solutions, not just a single point. This concept is fundamental in many areas of math, from algebra to calculus, and understanding how to manipulate and represent these solution sets is a key skill. We'll be using our knowledge of basic arithmetic operations, order of operations (PEMDAS/BODMAS, remember that?), and the properties of inequalities, like how adding or subtracting the same number from both sides doesn't change the inequality, and how multiplying or dividing by a positive number also keeps the inequality the same. We'll also touch upon what happens when we multiply or divide by a negative number – hint: it flips the inequality sign! This is super important and often a common tripping point for many, so we'll give it special attention. Our target inequality is: $2-2(y-1)<-5 y+10$. Let's get our hands dirty and start simplifying!

Step-by-Step Solution

Alright, fam, let's dive into solving our inequality: $2-2(y-1)<-5 y+10$. The first thing we want to do is simplify both sides of the inequality as much as possible. On the left side, we've got parentheses, so we need to use the distributive property. We'll multiply the -2 by everything inside the parentheses: -2 times y is $-2y$, and -2 times -1 is +2. So, the left side becomes $2 - 2y + 2$. Let's combine the constants on the left: 2 + 2 gives us 4. So, the simplified left side is $4 - 2y$. Now, let's look at the right side: $-5y + 10$. This side is already as simple as it can get. So, our inequality now looks like: $4 - 2y < -5y + 10$.

Our next goal is to get all the 'y' terms on one side and all the constant terms on the other side. I usually like to move the variable terms to the side that will result in a positive coefficient for the variable, if possible. Let's add $5y$ to both sides of the inequality. Remember, adding the same term to both sides doesn't change the direction of the inequality sign. So, we have: $4 - 2y + 5y < -5y + 10 + 5y$. Simplifying this gives us $4 + 3y < 10$.

Now, we want to isolate the term with 'y'. Let's subtract 4 from both sides: $4 + 3y - 4 < 10 - 4$. This simplifies to $3y < 6$.

Finally, to solve for 'y', we need to divide both sides by 3. Since 3 is a positive number, the inequality sign stays the same: rac{3y}{3} < rac{6}{3}. This gives us our solution: $y < 2$.

So, the solution to our inequality is that 'y' must be any number less than 2. Pretty straightforward when you break it down, right? We used the distributive property, combined like terms, and then isolated the variable using addition and division. Each step was designed to maintain the truth of the original statement. The distributive property helped us eliminate the parentheses, making the expression easier to manage. Combining like terms on each side (though only the left side had constants to combine initially) is a standard simplification technique. Moving the variable terms to one side and the constant terms to the other is crucial for isolating the variable. We chose to add $5y$ to both sides to get a positive coefficient for 'y', which often makes the final division step simpler. Subtracting 4 from both sides then isolated the $3y$ term. The final step of dividing by 3, a positive number, allowed us to solve for 'y' without changing the direction of the inequality. It's important to remember that if we had divided by a negative number, we would have had to flip the inequality sign. This careful execution of algebraic steps ensures that our final result, $y < 2$, accurately represents all values that satisfy the original inequality $2-2(y-1)<-5 y+10$.

Graphing the Solution Set

Now that we've found our solution, $y < 2$, let's bring it to life by graphing it on a number line. This is where things get visual, guys! Imagine a standard number line stretching infinitely in both directions, with 0 in the middle, positive numbers to the right, and negative numbers to the left. Our solution is $y < 2$, meaning 'y' can be any number that is strictly less than 2. This includes numbers like 1, 0, -1, -1.5, 1.999, and so on, all the way down to negative infinity. It does not include the number 2 itself because the inequality is strictly '<' and not '≤' (less than or equal to).

To represent this on a number line, we first find the number 2. We'll place a special mark, called an open circle, at the point representing 2. This open circle is super important because it signifies that the number 2 is not included in our solution set. If our inequality had been $y ext{ ≤ } 2$, we would have used a closed circle (a filled-in circle) to indicate that 2 is part of the solution. But for $y < 2$, it's an open circle.

Next, we need to show all the numbers less than 2. On a number line, numbers less than a certain point are always to the left of that point. So, we draw a bold line or an arrow extending from the open circle at 2 and going to the left, covering all the numbers stretching towards negative infinity. This shaded line visually represents the infinite set of numbers that satisfy our inequality.

Think of it like this: the open circle is a