Solving The Inequality 8z + 3 - 2z < 51 A Step By Step Guide

In this article, we will delve into the process of solving the linear inequality 8z + 3 - 2z < 51. Linear inequalities, a fundamental concept in algebra, are mathematical expressions that compare two values using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding how to solve these inequalities is crucial for various applications in mathematics, science, and engineering. We will break down the problem step-by-step, providing a clear and concise explanation of each step involved in isolating the variable 'z' and determining the solution set.

This detailed guide will not only help you understand the specific problem at hand but also equip you with the skills to tackle similar inequalities with confidence. We will cover the basic principles of inequality manipulation, including combining like terms, applying inverse operations, and handling potential sign changes when multiplying or dividing by a negative number. By the end of this article, you'll have a solid grasp of the techniques required to solve linear inequalities effectively.

Whether you're a student learning algebra for the first time or someone looking to refresh your knowledge, this article provides a comprehensive approach to solving linear inequalities. We will emphasize the importance of understanding the underlying concepts rather than simply memorizing steps, which will enable you to apply these skills to a wider range of problems. Let's embark on this journey of solving inequalities and unlock the power of algebraic problem-solving.

1. Combine Like Terms

The first step in solving the inequality 8z + 3 - 2z < 51 involves simplifying the expression by combining like terms. Like terms are terms that contain the same variable raised to the same power. In this case, we have two terms with the variable 'z': 8z and -2z. We can combine these terms by adding their coefficients. The coefficient of a term is the numerical factor that multiplies the variable.

When combining like terms, it's essential to pay attention to the signs of the coefficients. In our expression, we have a positive 8z and a negative -2z. To combine these terms, we perform the operation 8 - 2, which equals 6. Therefore, the combined term is 6z. The constant term +3 remains unchanged as there are no other constant terms to combine it with.

By combining like terms, we simplify the inequality from 8z + 3 - 2z < 51 to 6z + 3 < 51. This simplification makes the inequality easier to work with and brings us closer to isolating the variable 'z'. The principle of combining like terms is a fundamental aspect of algebraic manipulation, and it's crucial for solving equations and inequalities efficiently.

This step highlights the importance of recognizing and combining like terms as a fundamental step in simplifying algebraic expressions. By reducing the complexity of the inequality, we pave the way for the subsequent steps in isolating the variable and finding the solution set. The ability to combine like terms effectively is a key skill in algebra and is essential for tackling more complex problems.

2. Isolate the Variable Term

After combining like terms, the inequality is now in the form 6z + 3 < 51. The next crucial step is to isolate the variable term, which in this case is 6z. Isolating the variable term means getting it by itself on one side of the inequality. To achieve this, we need to eliminate the constant term +3 that is added to 6z.

To eliminate +3, we apply the inverse operation. The inverse operation of addition is subtraction. Therefore, we subtract 3 from both sides of the inequality. It's crucial to perform the same operation on both sides of the inequality to maintain the balance and ensure that the solution remains valid. Subtracting 3 from both sides gives us:

6z + 3 - 3 < 51 - 3

Simplifying this expression, we get:

6z < 48

By subtracting 3 from both sides, we have successfully isolated the variable term 6z on the left side of the inequality. This step is a critical part of the process as it brings us closer to isolating the variable 'z' itself. The principle of applying inverse operations to both sides of an equation or inequality is a fundamental concept in algebra and is used extensively in solving various types of problems.

This step demonstrates the importance of using inverse operations to isolate the variable term in an inequality. By carefully applying the subtraction operation to both sides, we maintain the balance of the inequality and move closer to the solution. Isolating the variable term is a key step in the process of solving for the variable and understanding the range of values that satisfy the inequality.

3. Solve for the Variable

Having isolated the variable term, our inequality now reads 6z < 48. The final step in solving for 'z' is to isolate the variable itself. Currently, 'z' is being multiplied by 6. To undo this multiplication and isolate 'z', we need to apply the inverse operation, which is division. We will divide both sides of the inequality by 6.

Dividing both sides by 6 ensures that we maintain the balance of the inequality. It's crucial to remember that when dividing or multiplying both sides of an inequality by a negative number, we must flip the direction of the inequality sign. However, in this case, we are dividing by a positive number (6), so the inequality sign remains unchanged.

Dividing both sides of 6z < 48 by 6, we get:

(6z) / 6 < 48 / 6

Simplifying this expression, we find:

z < 8

This is the solution to the inequality. It tells us that any value of 'z' that is less than 8 will satisfy the original inequality 8z + 3 - 2z < 51. The solution z < 8 represents an infinite set of values, all of which are less than 8. This highlights a key difference between solving equations and solving inequalities: equations typically have a finite number of solutions, while inequalities often have an infinite range of solutions.

This step showcases the final stage in solving the inequality, where we isolate the variable by applying the appropriate inverse operation. Dividing both sides by the coefficient of the variable allows us to determine the range of values that satisfy the inequality. The solution z < 8 provides a clear and concise answer to the problem, indicating that all values of 'z' less than 8 are solutions.

Therefore, the solution to the inequality 8z + 3 - 2z < 51 is z < 8. This corresponds to option D) z < 8.

In conclusion, solving linear inequalities involves a systematic approach of simplifying the expression, isolating the variable term, and then solving for the variable. By following these steps carefully and applying the principles of inverse operations and maintaining balance, we can effectively determine the solution set for any linear inequality. Understanding these concepts is fundamental for further studies in algebra and related fields.