Solving Inequalities: A Step-by-Step Guide For -4b ≤ -8

Let's dive into the world of inequalities! In this comprehensive guide, we'll tackle the inequality $-4b vertleq -8$ head-on. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), $vertleq$ (less than or equal to), and $vertgeq$ (greater than or equal to). Understanding how to solve them is crucial for various mathematical applications. Solving inequalities is a fundamental concept in mathematics, crucial for understanding various mathematical concepts and real-world applications. This guide provides a step-by-step approach to solving the inequality $-4b vertleq -8$, ensuring you grasp the process thoroughly. We will break down each step, providing clear explanations and insights to help you confidently tackle similar problems. So, grab your pencils, and let's get started on this mathematical journey!

Understanding Inequalities

Before we jump into solving our specific inequality, let's establish a firm grasp of what inequalities are and how they differ from equations. Inequalities are mathematical statements that compare two expressions using inequality symbols. Unlike equations, which state that two expressions are equal, inequalities indicate a range of possible values that satisfy the condition. The four primary inequality symbols are:

• < (less than): Indicates that the value on the left side is smaller than the value on the right side.
• > (greater than): Indicates that the value on the left side is larger than the value on the right side.
• $vertleq$ (less than or equal to): Indicates that the value on the left side is either smaller than or equal to the value on the right side.
• $vertgeq$ (greater than or equal to): Indicates that the value on the left side is either larger than or equal to the value on the right side.

Understanding these symbols is the first step in solving inequalities. Just like with equations, our goal is to isolate the variable to determine the range of values that make the inequality true. However, there's a crucial difference when dealing with inequalities: multiplying or dividing by a negative number requires flipping the inequality sign. This is a key concept we'll explore in detail as we solve our example. Recognizing the difference between equations and inequalities is essential. Equations have a single solution (or a finite set of solutions), while inequalities typically have a range of solutions. This range can be represented on a number line or expressed in interval notation. Grasping this difference is crucial for interpreting and applying the solutions we find. Inequalities play a vital role in various real-world scenarios, from determining budget constraints to optimizing resources. By understanding how to solve them, you gain a valuable tool for problem-solving in many areas of life. So, let's delve deeper into the specifics of solving $-4b vertleq -8$.

Step 1: Isolating the Variable

Our main goal in solving any inequality (or equation) is to isolate the variable. This means getting the variable (in our case, 'b') by itself on one side of the inequality. To do this, we use inverse operations – operations that undo each other. In the inequality $-4b vertleq -8$, 'b' is being multiplied by -4. The inverse operation of multiplication is division. Therefore, to isolate 'b', we need to divide both sides of the inequality by -4. This is where a critical rule comes into play: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This rule is essential for maintaining the truth of the inequality. Dividing by a negative number essentially reverses the relationship between the two sides of the inequality, hence the need to flip the sign. If we didn't flip the sign, we would end up with an incorrect solution set. Let's apply this to our inequality. We divide both sides of $-4b vertleq -8$ by -4. Remember, since we're dividing by a negative number, we must flip the $vertleq$ sign to $vertgeq$. This gives us: $b vertgeq 2$. This is a crucial step, and understanding why we flip the sign is essential for solving inequalities correctly. Imagine a number line: multiplying or dividing by a negative number reflects the numbers across zero, effectively changing their order. Therefore, flipping the inequality sign compensates for this reversal. So, with this crucial step completed, we've isolated our variable and are one step closer to the solution.

Step 2: Performing the Division

Now that we know we need to divide both sides of the inequality $-4b vertleq -8$ by -4 and flip the inequality sign, let's actually perform the division. Dividing both sides by -4, we get: $(-4b) / (-4) vertgeq (-8) / (-4)$. On the left side, -4 divided by -4 cancels out, leaving us with just 'b'. On the right side, -8 divided by -4 equals 2. So, our inequality now looks like this: $b vertgeq 2$. This result tells us that 'b' is greater than or equal to 2. This is a concise way of expressing the solution set for the inequality. It means that any value of 'b' that is 2 or larger will satisfy the original inequality, $-4b vertleq -8$. It's important to double-check our work to ensure we haven't made any errors. We can do this by plugging a value greater than or equal to 2 back into the original inequality. For instance, let's try b = 3: -4 * 3 = -12, and -12 is indeed less than or equal to -8. This confirms that our solution is likely correct. The division step is a straightforward arithmetic operation, but it's crucial to perform it accurately. A small mistake here can lead to an incorrect solution set. So, take your time, double-check your calculations, and ensure you're dividing both sides by the correct number, including the sign.

Step 3: Expressing the Solution

We've arrived at the solution: $b vertgeq 2$. This inequality tells us that the solution set includes all values of 'b' that are greater than or equal to 2. But how can we express this solution in a clear and comprehensive way? There are a few common methods: inequality notation (which we've already used), graphical representation on a number line, and interval notation. We've already expressed the solution using inequality notation: $b vertgeq 2$. This is a concise and direct way to state the solution. To represent the solution graphically, we can draw a number line. We mark the number 2 on the number line. Since our inequality includes