Solving Inequalities A Step-by-Step Guide To -4(2x + 1) ≤ 3(x - 5)

Navigating the world of inequalities is a fundamental skill in mathematics, with applications spanning various fields, from economics to engineering. This article provides a comprehensive guide on how to solve inequalities, focusing on the specific example of -4(2x + 1) ≤ 3(x - 5). We will break down the steps involved, ensuring a clear understanding of the underlying principles and techniques. By the end of this guide, you'll be well-equipped to tackle similar inequality problems with confidence.

Understanding Inequalities

Before diving into the solution, let's first understand what inequalities are. Unlike equations that show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. These relationships are represented by the symbols >, <, ≥, and ≤, respectively. In essence, solving an inequality means finding the range of values that satisfy the given relationship. This range is often represented graphically on a number line or in interval notation.

Solving inequalities shares many similarities with solving equations, but there's a crucial difference to keep in mind: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This is because multiplying by a negative number essentially flips the number line, changing the order of values. For example, if 2 < 4, then multiplying both sides by -1 gives -2 > -4. Failing to account for this rule is a common mistake when solving inequalities, so it's essential to be mindful of it.

In the context of the inequality -4(2x + 1) ≤ 3(x - 5), we aim to find all values of x that make the left-hand side less than or equal to the right-hand side. This involves a series of algebraic manipulations, guided by the same principles used in solving equations, but with careful attention to the rule about multiplying or dividing by negative numbers. Let's proceed step-by-step to unravel the solution.

Step-by-Step Solution

To solve the inequality -4(2x + 1) ≤ 3(x - 5), we will follow a systematic approach, similar to solving equations, but with careful attention to the rules governing inequalities. The key is to isolate the variable x on one side of the inequality while maintaining the truth of the relationship. This involves simplifying both sides, combining like terms, and strategically applying inverse operations.

1. Distribute

The first step is to eliminate the parentheses by applying the distributive property. This means multiplying the terms outside the parentheses by each term inside the parentheses. On the left-hand side, we multiply -4 by both 2x and 1, and on the right-hand side, we multiply 3 by both x and -5. This gives us:

-4 * (2x) + (-4) * 1 ≤ 3 * x + 3 * (-5)

Simplifying this, we get:

-8x - 4 ≤ 3x - 15

Now the inequality is free of parentheses, making it easier to manipulate and isolate the variable x. This step is crucial as it sets the stage for the subsequent steps of combining like terms and applying inverse operations.

2. Combine Like Terms

Next, we want to gather all the terms involving x on one side of the inequality and all the constant terms on the other side. This is achieved by adding or subtracting terms from both sides of the inequality. To get all the x terms on the right-hand side, we add 8x to both sides:

-8x - 4 + 8x ≤ 3x - 15 + 8x

Simplifying, we get:

-4 ≤ 11x - 15

Now, to isolate the x term further, we add 15 to both sides to get all the constant terms on the left-hand side:

-4 + 15 ≤ 11x - 15 + 15

Simplifying again, we have:

11 ≤ 11x

At this point, we have successfully grouped the x terms and the constant terms on opposite sides of the inequality. The next step is to isolate x completely by dividing both sides by its coefficient.

3. Isolate the Variable

To finally isolate x, we need to divide both sides of the inequality by the coefficient of x, which is 11. Since 11 is a positive number, we do not need to reverse the direction of the inequality sign. Dividing both sides by 11, we get:

11 / 11 ≤ 11x / 11

Simplifying, we obtain:

1 ≤ x

This can also be written as:

x ≥ 1

This is the solution to the inequality. It states that x can be any value greater than or equal to 1. This solution represents a range of values, rather than a single value as in an equation. To fully understand the solution, it's helpful to visualize it on a number line or express it in interval notation.

Interpreting the Solution

The solution to the inequality -4(2x + 1) ≤ 3(x - 5) is x ≥ 1. This means that any value of x that is greater than or equal to 1 will satisfy the original inequality. We can visualize this solution on a number line by shading the region to the right of 1, including 1 itself, which is indicated by a closed circle or bracket at 1.

Alternatively, we can express the solution in interval notation. Interval notation is a concise way of representing a set of numbers using intervals. The solution x ≥ 1 can be written in interval notation as [1, ∞). The square bracket at 1 indicates that 1 is included in the solution set, while the parenthesis at ∞ indicates that infinity is not a specific number and is not included in the set.

To further solidify our understanding, we can test some values. Let's try x = 1, x = 2, and x = 0:

• If x = 1:

  -4(2(1) + 1) ≤ 3(1 - 5)

  -4(3) ≤ 3(-4)

  -12 ≤ -12 (True)

• If x = 2:

  -4(2(2) + 1) ≤ 3(2 - 5)

  -4(5) ≤ 3(-3)

  -20 ≤ -9 (True)

• If x = 0:

  -4(2(0) + 1) ≤ 3(0 - 5)

  -4(1) ≤ 3(-5)

  -4 ≤ -15 (False)

As we can see, values greater than or equal to 1 satisfy the inequality, while values less than 1 do not. This confirms our solution and demonstrates the importance of understanding the range of values that satisfy an inequality.

Common Mistakes to Avoid

Solving inequalities, while sharing similarities with solving equations, has its own set of potential pitfalls. Being aware of these common mistakes can help you avoid errors and arrive at the correct solution. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. As we discussed earlier, this is a crucial rule to remember, as it stems from the way negative numbers affect the order of values on the number line. Always double-check this step whenever you perform multiplication or division by a negative number.

Another common mistake is incorrectly applying the distributive property. Make sure to distribute the term outside the parentheses to every term inside the parentheses. Pay close attention to the signs, especially when dealing with negative numbers. A simple sign error can lead to a completely different solution.

Furthermore, be careful when combining like terms. Ensure that you are only combining terms that have the same variable and exponent. For example, you can combine 3x and 5x, but you cannot combine 3x and 5x². Also, remember to perform the same operation on both sides of the inequality to maintain the balance and preserve the truth of the relationship.

Finally, always check your solution by substituting values from the solution set back into the original inequality. This helps you verify that your solution is correct and that you haven't made any errors along the way. By being mindful of these common mistakes and taking the time to check your work, you can significantly improve your accuracy in solving inequalities.

Conclusion

In this comprehensive guide, we have explored the process of solving the inequality -4(2x + 1) ≤ 3(x - 5). We started by understanding the basic principles of inequalities and the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number. We then systematically worked through the steps of distributing, combining like terms, and isolating the variable, arriving at the solution x ≥ 1. We interpreted this solution graphically on a number line and expressed it in interval notation as [1, ∞). Finally, we discussed common mistakes to avoid, emphasizing the importance of careful attention to detail and checking your work.

Mastering the techniques for solving inequalities is an essential skill in mathematics, with applications in various fields. By understanding the underlying principles and practicing regularly, you can build confidence and proficiency in tackling inequality problems. Remember to always think critically, pay attention to the details, and check your solutions. With these strategies, you'll be well-equipped to navigate the world of inequalities and solve them with accuracy and ease.