Solving Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of inequalities and learn how to solve them and represent the solutions graphically. We'll tackle the inequality $16 > 7(t - 1) - 5$ together. Don't worry, it's not as scary as it looks. We'll break it down into easy-to-follow steps. This guide will walk you through the process, ensuring you understand each stage. Ready to get started? Let's go!

Understanding the Basics of Inequalities

First things first, what exactly are inequalities? Well, unlike equations that use an equals sign (=), inequalities use symbols like greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). These symbols show a relationship between two values where one is not necessarily equal to the other. Solving an inequality means finding the range of values that make the inequality true. The solutions aren't just one specific number, but rather a whole set of numbers! We'll discover how to pinpoint these solutions and visualize them on a number line.

Now, before we jump into the specific problem, let's brush up on a few key concepts. Remember that when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is super important to remember, or you'll get the wrong answer! Think of it like this: if you have a number line, and you're moving from left to right, the numbers get bigger. If you multiply by -1, you're essentially flipping the number line, and the direction of the inequality changes to accommodate this flip. Also, we'll need to know about the distributive property (a(b + c) = ab + ac) since it appears in our example. This property is used to expand expressions by multiplying a term outside the parentheses with each term inside the parentheses. Another handy thing to keep in mind is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Although the order of operations primarily applies when simplifying a single expression, it helps to keep our calculations organized and accurate when solving inequalities. Finally, when graphing, we'll use open circles to represent values that aren't included in the solution (for < and >) and closed circles to represent values that are included (for ≤ and ≥).

Step 1: Simplify the Inequality

Let's start by looking at our inequality: $16 > 7(t - 1) - 5$. The first thing we need to do is simplify it. We'll use the distributive property to get rid of the parentheses. That means we multiply the 7 by both terms inside the parentheses. So, $7 * t$ gives us $7t$, and $7 * -1$ gives us $-7$. Now the inequality looks like this: $16 > 7t - 7 - 5$. Next, we'll combine the constant terms on the right side of the inequality. We have $-7 - 5$, which simplifies to $-12$. Our inequality is now: $16 > 7t - 12$. See? We're already making progress!

Isolating the Variable

Now that we have a simplified inequality, we need to isolate the variable, 't'. This means getting 't' by itself on one side of the inequality. To do this, we'll use inverse operations. Remember that whatever we do to one side of the inequality, we must do to the other side to keep it balanced.

Step 2: Adding to Both Sides

Our inequality is currently $16 > 7t - 12$. To get rid of the -12 on the right side, we'll add 12 to both sides of the inequality. Adding 12 to the left side gives us $16 + 12 = 28$. On the right side, $-12 + 12$ cancels out, leaving us with just $7t$. Now, the inequality looks like this: $28 > 7t$. See how we're slowly but surely getting closer to solving for 't'?

Step 3: Dividing to Isolate the Variable

Almost there, guys! We have $28 > 7t$, and we need to isolate 't'. Right now, 't' is being multiplied by 7. To undo that, we'll divide both sides of the inequality by 7. Dividing 28 by 7 gives us 4. Dividing $7t$ by 7 gives us 't'. So, we end up with: $4 > t$. Or, we can rewrite it as $t < 4$. This tells us that any value of 't' that is less than 4 will make the original inequality true. We have solved the inequality!

Graphing the Solution

Now, let's visualize our solution on a number line. Graphing inequalities is super easy! Here's how to do it. First, draw a number line. Mark the number 4 on the number line. Since our inequality is $t < 4$, we're looking for all numbers less than 4. This means we'll place an open circle (because 4 itself is not included) at the point 4 on the number line. Then, we'll draw a line going to the left from the open circle. This line represents all the numbers that are less than 4. So, any number to the left of 4 on the number line, such as 3, 2, 1, 0, and even negative numbers, will satisfy the inequality $t < 4$. The graph clearly shows the range of values for 't' that make the inequality true. The open circle indicates that 4 itself is not included in the solution set.

Graphing Steps

To summarize the graphing process:

1. Draw a Number Line: Start by drawing a straight line and marking some key numbers on it, like 0, 4, and a few numbers on either side of 4 (e.g., 2, 3, 5, 6).
2. Mark the Critical Point: Locate the number 4 on the number line. This is the critical point in our solution.
3. Choose Circle Type: Since the inequality is $t < 4$ (less than), we use an open circle at the point 4. This indicates that 4 is not included in the solution.
4. Draw the Arrow: Draw an arrow from the open circle at 4, extending to the left. This arrow represents all the numbers less than 4. The arrow indicates the direction of all the values of t that make the inequality true.

Checking the Solution

It's always a good idea to check your solution to make sure you're on the right track! Let's pick a number that's less than 4, like 0, and plug it back into our original inequality, $16 > 7(t - 1) - 5$. Substituting 0 for 't', we get: $16 > 7(0 - 1) - 5$. Simplifying, we get: $16 > 7(-1) - 5$, then $16 > -7 - 5$, and finally, $16 > -12$. This statement is true! Since 16 is indeed greater than -12, our solution, $t < 4$, seems correct. Let's also check a number greater than 4, like 5. Substituting 5 for 't', we get: $16 > 7(5 - 1) - 5$, then $16 > 7(4) - 5$, and $16 > 28 - 5$, which simplifies to $16 > 23$. This is false! This confirms that our solution $t < 4$ is accurate because only values less than 4 satisfy the inequality.

Conclusion

Alright, guys! We've successfully solved the inequality $16 > 7(t - 1) - 5$ and graphed the solution. We found that $t < 4$, meaning any number less than 4 makes the inequality true. Remember the key steps: simplify, isolate the variable, and graph the solution on a number line. Keep practicing, and you'll become a pro at solving inequalities in no time! Keep in mind to always check your solution by substituting a value within the solution set and one outside the solution set back into the original inequality. This helps to confirm your answer. If you get stuck, don't worry! Go back through the steps, review the concepts, and don't hesitate to ask for help. Happy solving! Hopefully, you all have a better understanding of how to solve and graph inequalities! This is a fundamental concept in mathematics. Keep practicing, and you'll get the hang of it! Remember, mathematics is all about practice and understanding the underlying concepts. So keep at it, and you'll see great results!