Solve This Inequality 8z+3-2z 51

Solving the Inequality: 8z + 3 - 2z ≤ 51 – A Step-by-Step Guide

This article provides a practical guide on how to solve the inequality 8z + 3 - 2z ≤ 51. We'll break down the process step-by-step, explaining the underlying mathematical principles, and offering additional insights to help you master solving linear inequalities. Understanding inequalities is crucial in various fields, from basic algebra to advanced calculus and even real-world problem-solving. This guide is designed for learners of all levels, from beginners grappling with the basics to those seeking a refresher or deeper understanding.

Introduction: Understanding Inequalities

Unlike equations, which focus on finding a specific value that makes the statement true, inequalities deal with a range of values. Solving inequalities involves isolating the variable (in this case, 'z') to find the set of values that satisfy the inequality. On the flip side, the inequality symbol "≤" (less than or equal to) indicates that the expression on the left-hand side is less than or equal to the expression on the right-hand side. The process is similar to solving equations, but with one key difference: when you multiply or divide by a negative number, you must reverse the inequality sign.

Step-by-Step Solution: 8z + 3 - 2z ≤ 51

Let's tackle the inequality step-by-step:

1. Combine Like Terms: The first step is to simplify the left-hand side of the inequality by combining like terms. We have 8z and -2z, which are both terms containing the variable 'z'. Combining them gives us:

   6z + 3 ≤ 51

2. Isolate the Variable Term: Our goal is to isolate the term with 'z'. To do this, we need to get rid of the constant term (+3) on the left-hand side. We can achieve this by subtracting 3 from both sides of the inequality. Remember, whatever operation you perform on one side, you must perform on the other to maintain the balance of the inequality:

   6z + 3 - 3 ≤ 51 - 3

   6z ≤ 48

3. Solve for the Variable: Now, we need to isolate 'z' completely. The 'z' is currently being multiplied by 6. To undo this multiplication, we divide both sides of the inequality by 6:

   6z / 6 ≤ 48 / 6

   z ≤ 8

Because of this, the solution to the inequality 8z + 3 - 2z ≤ 51 is z ≤ 8. Basically, any value of 'z' that is less than or equal to 8 will satisfy the original inequality.

Graphical Representation of the Solution

The solution z ≤ 8 can be represented graphically on a number line. Worth adding: this shaded region represents all the values of 'z' that satisfy the inequality. In real terms, you would draw a number line, mark the point 8, and shade the region to the left of 8, including 8 itself (because it's "less than or equal to"). A closed circle (or a filled-in dot) is used at 8 to indicate that 8 is included in the solution set.

Understanding the Solution Set

The solution z ≤ 8 represents an infinite number of solutions. Any real number less than or equal to 8 will satisfy the inequality. For example:

• z = 8 is a solution (8 ≤ 8 is true)
• z = 0 is a solution (0 ≤ 8 is true)
• z = -5 is a solution (-5 ≤ 8 is true)
• z = 7.999 is a solution (7.999 ≤ 8 is true)

Even so, values like z = 9, z = 10, or any number greater than 8 will not satisfy the inequality.

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Explanation of Mathematical Principles

The solution process relies on several fundamental mathematical principles:

• The Addition Property of Inequality: If you add the same number to both sides of an inequality, the inequality remains true.
• The Subtraction Property of Inequality: If you subtract the same number from both sides of an inequality, the inequality remains true.
• The Multiplication Property of Inequality: If you multiply both sides of an inequality by the same positive number, the inequality remains true. Still, if you multiply by a negative number, you must reverse the inequality sign.
• The Division Property of Inequality: Similar to multiplication, dividing both sides by a positive number preserves the inequality, while dividing by a negative number requires reversing the inequality sign. These properties confirm that the solution set we find accurately reflects all the values that satisfy the original inequality.

Frequently Asked Questions (FAQ)

• Q: What if the inequality symbol was > (greater than) instead of ≤?

  A: If the inequality was 8z + 3 - 2z > 51, the steps would be the same until the final step. The solution would then be z > 8. The graphical representation would involve shading the region to the right of 8, using an open circle (or an unfilled dot) at 8 to show that 8 is not included in the solution set.

• Q: What happens if there are fractions or decimals in the inequality?

  A: The process remains the same. You would still combine like terms, isolate the variable, and solve for the variable using the properties of inequalities. You may need to use fractions or decimals in your calculations. Still holds up.

• Q: Can I check my answer?

  A: Yes! Choose a value from your solution set (e.If the inequality holds true, then your solution is correct. Practically speaking, g. g.You can also test a value outside your solution set (e., z = 0 or z = 8) and substitute it back into the original inequality. , z = 9) to verify that it does not satisfy the inequality.

• Q: What if the inequality involves more than one variable?

  A: Solving inequalities with more than one variable involves different techniques, often leading to solutions represented as regions on a coordinate plane rather than a single number line. These techniques are typically covered in more advanced algebra courses.

Conclusion: Mastering Inequalities

Solving inequalities, like the example 8z + 3 - 2z ≤ 51, is a fundamental skill in algebra. Remember to pay close attention to the inequality symbol and reverse it when multiplying or dividing by a negative number. Worth adding: practice is key to mastering this skill; work through various examples, and don't hesitate to seek further guidance if needed. With consistent effort and a grasp of the underlying principles, solving inequalities will become second nature. By understanding the properties of inequalities and following the step-by-step process, you can confidently solve a wide range of inequalities. This full breakdown aims to equip you with the knowledge and confidence to tackle such problems effectively. The ability to solve inequalities opens doors to a deeper understanding of mathematical concepts and their application in various fields. Remember to always check your solution by substituting values into the original inequality to verify the accuracy of your answer.