Solve The Inequality 8z+3-2z 51

Solving the Inequality: 8z + 3 - 2z ≤ 51 – A complete walkthrough

This article provides a full breakdown on how to solve the linear inequality 8z + 3 - 2z ≤ 51. We'll break down the process step-by-step, explaining the underlying principles of inequality solving and offering additional tips and examples to solidify your understanding. That's why mastering this type of problem is crucial for success in algebra and beyond. Understanding inequalities is fundamental in various fields, from computer science to economics, enabling us to model and solve real-world problems involving constraints and limitations.

Introduction to Inequalities

Before diving into the solution, let's clarify the concept of inequalities. Unlike equations, which state that two expressions are equal (=), inequalities show a relationship of order between two expressions. The main inequality symbols are:

• ≤ (less than or equal to)
• < (less than)
• ≥ (greater than or equal to)
• > (greater than)

Solving an inequality involves finding the range of values for the variable (in this case, 'z') that make the inequality true. And the solution is typically expressed as an interval or a set of values. Remember that when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign.

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

Now, let's tackle the inequality 8z + 3 - 2z ≤ 51. We'll follow these steps:

1. Combine Like Terms:

The first step is to simplify the left side of the inequality by combining like terms. We have two terms with 'z': 8z and -2z. Adding these together, we get:

8z - 2z = 6z

Our inequality now becomes:

6z + 3 ≤ 51

2. Isolate the Variable Term:

Next, we want to isolate the term containing 'z' (6z). To do this, we subtract 3 from both sides of the inequality:

6z + 3 - 3 ≤ 51 - 3

This simplifies to:

6z ≤ 48

3. Solve for the Variable:

Finally, we solve for 'z' by dividing both sides of the inequality by 6:

6z / 6 ≤ 48 / 6

This gives us:

z ≤ 8

That's why, the solution to the inequality 8z + 3 - 2z ≤ 51 is z ≤ 8. Basically, any value of 'z' less than or equal to 8 will satisfy the inequality.

Representing the Solution

The solution, z ≤ 8, can be represented in several ways:

• Interval Notation: (-∞, 8] This notation indicates that the solution includes all real numbers from negative infinity up to and including 8. The square bracket '[' indicates that 8 is included in the solution set.

• Number Line: A number line can visually represent the solution. You would draw a closed circle at 8 (because 8 is included) and shade the line to the left of 8, extending towards negative infinity.

• Set-Builder Notation: {z | z ∈ ℝ, z ≤ 8} This notation reads as "the set of all z such that z is a real number and z is less than or equal to 8."

Explanation of the Underlying Principles

The steps we followed are based on the properties of inequalities. These properties make it possible to manipulate inequalities while preserving the truth of the inequality statement. Here's a breakdown:

• Addition Property of Inequality: If a ≤ b, then a + c ≤ b + c for any real number c. This means you can add or subtract the same number from both sides of an inequality without changing the direction of the inequality sign.

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• Multiplication Property of Inequality: If a ≤ b and c > 0, then ac ≤ bc. If a ≤ b and c < 0, then ac ≥ bc. This means you can multiply or divide both sides of an inequality by a positive number without changing the direction of the inequality sign. That said, if you multiply or divide by a negative number, you must reverse the direction of the inequality sign.

These properties are fundamental to solving any linear inequality. Understanding them will allow you to tackle more complex inequalities with confidence.

Further Examples and Practice Problems

Let's work through a few more examples to solidify your understanding:

Example 1: 5x - 7 > 18

1. Add 7 to both sides: 5x > 25
2. Divide both sides by 5: x > 5

Example 2: -3y + 2 ≤ 11

1. Subtract 2 from both sides: -3y ≤ 9
2. Divide both sides by -3 (remember to reverse the inequality sign!): y ≥ -3

Example 3: 2(a + 4) < 6a - 10

1. Distribute the 2: 2a + 8 < 6a - 10
2. Subtract 2a from both sides: 8 < 4a - 10
3. Add 10 to both sides: 18 < 4a
4. Divide both sides by 4: 4.5 < a or a > 4.5

These examples demonstrate the application of the properties of inequalities in different scenarios. Remember to always carefully apply the rules of combining like terms and manipulating inequalities to ensure accurate solutions.

Frequently Asked Questions (FAQ)

Q1: What happens if I multiply or divide by a negative number?

A1: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. Here's one way to look at it: if you have -2x ≤ 6, dividing by -2 gives x ≥ -3.

Q2: Can I solve inequalities graphically?

A2: Yes, you can solve inequalities graphically. For linear inequalities, you would graph the corresponding equation and then determine which region of the coordinate plane satisfies the inequality.

Q3: What if the inequality involves fractions?

A3: Inequalities with fractions can be solved by first finding a common denominator and combining the fractions. Then, follow the same steps as outlined above.

Q4: What if the inequality is a compound inequality (e.g., 2 < x < 5)?

A4: Compound inequalities are solved by working with each part of the inequality separately. The solution will be the intersection of the solutions to each part.

Q5: How do I check my solution?

A5: To check your solution, substitute a value from the solution set into the original inequality. If the inequality is true, your solution is correct. As an example, if you found z ≤ 8, try substituting z = 7 or z = 8 into the original inequality 8z + 3 - 2z ≤ 51. If the inequality holds true for values within the solution set and false for values outside, then your solution is correct.

Conclusion

Solving inequalities, particularly linear inequalities like 8z + 3 - 2z ≤ 51, is a fundamental skill in algebra and mathematics in general. Also, practice is key to mastering this skill, so work through additional examples and put to use different representation methods to deepen your understanding and build confidence in your problem-solving abilities. By understanding the properties of inequalities and following the step-by-step process outlined above, you can confidently solve a wide range of inequality problems. Remember to pay close attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. With consistent practice and a firm grasp of the underlying principles, you'll be well-equipped to tackle more complex mathematical challenges.