Solving The Inequality -x/7 + 4 >= 3x A Step-by-Step Guide

Hey guys! Today, we're diving deep into solving a linear inequality. Inequalities might seem a bit tricky at first, but trust me, with a step-by-step approach, you'll be solving them like a pro in no time. We're going to tackle the inequality ${\frac{-x}{7} + 4 \geq 3x}$ and break down each step so you can understand exactly how to arrive at the correct solution. So, grab your pencils and paper, and let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are all about. Unlike equations, which show that two expressions are equal, inequalities show relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols we use for these relationships are: > (greater than), < (less than), ${\geq}$ (greater than or equal to), and ${\leq}$ (less than or equal to). When we solve an inequality, we're finding the range of values that make the inequality true. This range can be a set of numbers, rather than just a single solution like in equations. Remember, the goal is to isolate the variable on one side of the inequality to determine this range. Now that we have the basics down, let's dive into the specific inequality we have at hand: ${\frac{-x}{7} + 4 \geq 3x}$. Our mission is to find all the values of x that satisfy this inequality. This involves a series of algebraic manipulations, keeping in mind that some operations, like multiplying or dividing by a negative number, will flip the direction of the inequality sign. We'll go through each step methodically, ensuring clarity and accuracy in our solution.

Step 1: Clearing the Fraction

The first thing we want to do is get rid of that fraction, because fractions can sometimes make things look more complicated than they really are. Our inequality is ${\frac{-x}{7} + 4 \geq 3x}$. To eliminate the fraction, we're going to multiply every single term in the inequality by the denominator, which in this case is 7. This is a crucial step because it simplifies the inequality, making it easier to work with. When we multiply, we need to ensure that every term, including the constants and the terms with x, gets multiplied by 7. This maintains the balance of the inequality. So, let's do it! Multiplying each term by 7 gives us: 7 * (${\frac{-x}{7}}$) + 7 * 4 ${\geq}$ 7 * (3x). Now, let's simplify this. The 7 in the denominator of the first term cancels out with the 7 we're multiplying by, leaving us with -x. Then, 7 * 4 equals 28, and 7 * 3x equals 21x. So, our inequality now looks like this: -x + 28 ${\geq}$ 21x. See how much cleaner that looks already? Getting rid of fractions is often the first step in solving inequalities, as it sets the stage for easier manipulation and ultimately, finding the solution. Now that we've cleared the fraction, we're one step closer to isolating x and figuring out the range of values that satisfy the original inequality. The next step involves gathering the x terms on one side and the constants on the other, which we'll tackle in the following section.

Step 2: Gathering the 'x' Terms

Alright, now that we've cleared the fraction, let's gather all the terms with 'x' on one side of the inequality. Our inequality currently looks like this: -x + 28 ${\geq}$ 21x. To get all the 'x' terms together, we can add 'x' to both sides of the inequality. Remember, whatever operation we perform on one side, we must also perform on the other side to maintain the balance. Adding 'x' to both sides is a strategic move because it eliminates the '-x' term on the left side, making the 'x' terms appear only on the right side. This helps in isolating 'x' and eventually solving for its possible values. So, let's go ahead and add 'x' to both sides: -x + x + 28 ${\geq}$ 21x + x. Simplifying this, we get: 28 ${\geq}$ 22x. Notice how we've successfully moved all the 'x' terms to the right side of the inequality. This is a crucial step in isolating 'x'. Now, we have a much simpler inequality to work with. The next step involves isolating 'x' completely by getting rid of the coefficient (the number multiplying 'x'). We'll do this by dividing both sides of the inequality by the coefficient. But before we do that, let's take a moment to appreciate how far we've come. We started with a slightly intimidating inequality with a fraction, and now we have a much cleaner expression. This illustrates the power of step-by-step problem-solving in mathematics. In the next section, we'll finalize the isolation of 'x' and determine the solution to our inequality.

Step 3: Isolating 'x'

We're almost there, guys! We've successfully cleared the fraction and gathered the 'x' terms on one side. Our inequality now reads: 28 ${\geq}$ 22x. To completely isolate 'x', we need to get rid of the coefficient 22. The way we do this is by dividing both sides of the inequality by 22. Remember, the golden rule of inequalities (and equations) is that whatever you do to one side, you must do to the other to maintain the balance. Dividing both sides by 22 will leave 'x' all by itself on the right side, which is exactly what we want. So, let's divide: ${\frac{28}{22}}$ ${\geq}$ ${\frac{22x}{22}}$. Now, let's simplify this. On the right side, 22 divided by 22 is simply 1, leaving us with 'x'. On the left side, ${\frac{28}{22}}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us ${\frac{14}{11}}$. So, our inequality now looks like this: ${\frac{14}{11}}$ ${\geq}$ x. We've done it! We've successfully isolated 'x'. This inequality tells us that ${\frac{14}{11}}$ is greater than or equal to x, which is the same as saying x is less than or equal to ${\frac{14}{11}}$. In mathematical notation, this is written as x ${\leq}$ ${\frac{14}{11}}$. This is our solution! It means that any value of x that is less than or equal to ${\frac{14}{11}}$ will satisfy the original inequality. In the next section, we'll summarize our steps and choose the correct answer from the given options.

Step 4: The Solution

Let's recap what we've done so far. We started with the inequality ${\frac{-x}{7} + 4 \geq 3x}$, and we wanted to find all the values of 'x' that make this inequality true. We went through a series of steps:

1. Cleared the fraction by multiplying every term by 7, which gave us -x + 28 ${\geq}$ 21x.
2. Gathered the 'x' terms by adding 'x' to both sides, resulting in 28 ${\geq}$ 22x.
3. Isolated 'x' by dividing both sides by 22 and simplifying, which led us to ${\frac{14}{11}}$ ${\geq}$ x, or x ${\leq}$ ${\frac{14}{11}}$.

So, our solution is x ${\leq}$ ${\frac{14}{11}}$. Now, let's look at the options given:

A) x ${\geq}$ -1 B) x ${\leq}$ ${\frac{2}{1}}$ C) x ${\leq}$ ${\frac{14}{11}}$ D) x ${\leq}$ ${\frac{2}{5}}$

Comparing our solution to the options, we can see that option C, x ${\leq}$ ${\frac{14}{11}}$, matches our result perfectly. Therefore, the correct answer is C. We've successfully solved the inequality! Remember, the key to solving inequalities is to follow a step-by-step approach, just like we did here. Clear fractions, gather like terms, and isolate the variable. And always remember to flip the inequality sign if you multiply or divide by a negative number. With practice, you'll become more and more confident in solving inequalities. In the final section, we'll discuss some common mistakes to avoid when solving inequalities.

Common Mistakes to Avoid

When solving inequalities, there are a few common pitfalls that students sometimes fall into. Being aware of these mistakes can help you avoid them and ensure you get the correct solution every time. One of the most crucial things to remember is: when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have -2x > 4, and you divide both sides by -2, the inequality sign changes from '>' to '<', giving you x < -2. Forgetting to flip the sign is a very common mistake and can lead to the wrong answer. Another mistake is not applying an operation to every term in the inequality. Just like in equations, whatever you do to one side, you must do to the entire other side. This means if you're multiplying by a number, you need to multiply every term on both sides by that number, not just some of them. Similarly, when clearing fractions, make sure you multiply every single term by the common denominator. A third common error is making arithmetic mistakes while simplifying the inequality. This could be anything from adding or subtracting numbers incorrectly to making errors when multiplying or dividing. To minimize these mistakes, it's always a good idea to double-check your work and perform each step carefully. Slowing down and being meticulous can save you from making simple errors that can change the entire outcome. Finally, some students get confused about the direction of the inequality sign and how to interpret the solution. Remember, x < a means x is less than a, and x > a means x is greater than a. x ${\leq}$ a means x is less than or equal to a, and x ${\geq}$ a means x is greater than or equal to a. Understanding these symbols and what they represent is crucial for correctly interpreting your solution. By keeping these common mistakes in mind and being careful with your steps, you can confidently solve inequalities and avoid these pitfalls. Happy solving!