Solving -8t + 2 > -75 Finding The Solutions To The Inequality

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In the realm of mathematics, inequalities play a crucial role in describing relationships between values that are not necessarily equal. Understanding how to solve inequalities is fundamental for various applications, ranging from optimizing real-world scenarios to analyzing mathematical models. In this article, we will delve into the process of solving the inequality -8t + 2 > -75, providing a step-by-step guide and exploring the solutions among the given options. This exploration will not only enhance your problem-solving skills but also provide a deeper understanding of how inequalities work.

Understanding Inequalities

Inequalities are mathematical statements that compare two expressions using symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike equations, which assert the equality of two expressions, inequalities define a range of values that satisfy a particular condition. Solving an inequality involves finding all values of the variable that make the inequality true. This process often requires algebraic manipulations similar to those used in solving equations, but with some key differences, especially when dealing with negative coefficients.

Solving the Inequality -8t + 2 > -75

The given inequality is -8t + 2 > -75. To solve for t, we need to isolate the variable on one side of the inequality. Here’s a detailed breakdown of the steps involved:

  1. Isolate the term with the variable:

    The first step is to isolate the term containing the variable, which in this case is -8t. To do this, we subtract 2 from both sides of the inequality:

    -8t + 2 - 2 > -75 - 2

    -8t > -77

  2. Divide by the coefficient of the variable:

    Now, we need to divide both sides by the coefficient of t, which is -8. It’s crucial to remember that when we divide or multiply both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. This is a fundamental rule in solving inequalities.

    (-8t) / -8 < (-77) / -8

    t < 77/8

  3. Simplify the result:

    The fraction 77/8 can be expressed as a mixed number or a decimal to better understand the solution range. In this case, 77/8 is equal to 9.625.

    t < 9.625

Thus, the solution to the inequality -8t + 2 > -75 is t < 9.625. This means any value of t that is less than 9.625 will satisfy the original inequality.

Testing the Given Solutions

Now that we have found the solution set for the inequality, we can test the given values to see which ones satisfy the condition t < 9.625. The values provided are t = 4, t = -2, t = -1, and t = -6. We will evaluate each value to determine if it falls within the solution set.

1. Testing t = 4

To test t = 4, we substitute this value into the inequality t < 9.625:

4 < 9.625

This statement is true, as 4 is indeed less than 9.625. Therefore, t = 4 is a solution to the inequality.

2. Testing t = -2

Next, we test t = -2 by substituting it into the inequality:

-2 < 9.625

This statement is also true, since -2 is less than 9.625. Thus, t = -2 is a solution.

3. Testing t = -1

Substituting t = -1 into the inequality, we get:

-1 < 9.625

This is true, as -1 is less than 9.625. Therefore, t = -1 is a solution.

4. Testing t = -6

Finally, we test t = -6:

-6 < 9.625

This statement is true, as -6 is less than 9.625. Thus, t = -6 is a solution.

Identifying the Solutions

After testing each value, we have found that t = 4, t = -2, t = -1, and t = -6 all satisfy the inequality -8t + 2 > -75. This comprehensive evaluation confirms that all the provided values are solutions to the inequality.

Graphical Representation of the Solution

Visualizing the solution on a number line can provide a clearer understanding of the inequality. The solution t < 9.625 represents all the values to the left of 9.625 on the number line. Since the inequality is strict (i.e., t is strictly less than 9.625), we use an open circle at 9.625 to indicate that this value is not included in the solution set. The values 4, -2, -1, and -6 are all located to the left of 9.625, confirming their status as solutions.

Why Reversing the Inequality Sign Matters

The rule of reversing the inequality sign when multiplying or dividing by a negative number is critical to understand. To illustrate why this is necessary, consider a simple inequality:

2 < 4

This statement is clearly true. Now, let’s multiply both sides by -1 without reversing the inequality sign:

-2 < -4

This statement is false, as -2 is greater than -4. However, if we reverse the inequality sign, we get:

-2 > -4

This statement is true. This example demonstrates that reversing the inequality sign is essential to maintain the validity of the inequality when dealing with negative numbers. In the context of our original inequality, dividing by -8 necessitated reversing the > sign to < to accurately reflect the solution set.

Common Mistakes to Avoid

When solving inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and improve your accuracy.

  1. Forgetting to Reverse the Inequality Sign: As discussed, failing to reverse the inequality sign when multiplying or dividing by a negative number is a frequent error. Always double-check this step when working with negative coefficients.

  2. Incorrectly Applying the Distributive Property: If the inequality involves parentheses, ensure you correctly apply the distributive property. For example, if you have an expression like -2(x + 3) > 4, distribute the -2 to both terms inside the parentheses: -2x - 6 > 4. Failing to distribute correctly can lead to an incorrect inequality.

  3. Arithmetic Errors: Simple arithmetic mistakes, such as adding or subtracting numbers incorrectly, can also lead to wrong solutions. Double-check your calculations to ensure accuracy.

  4. Misunderstanding Solution Sets: It’s important to understand that the solution to an inequality is often a range of values, not just a single value. Visualizing the solution on a number line can help you grasp the concept of a solution set.

Practical Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous practical applications in various fields. Understanding inequalities can help in real-world problem-solving and decision-making.

  1. Optimization Problems: Inequalities are frequently used in optimization problems, where the goal is to maximize or minimize a certain quantity subject to constraints. For example, a business might want to maximize profit while staying within a budget or production capacity.

  2. Resource Allocation: Inequalities can help determine how to allocate resources effectively. For instance, a project manager might use inequalities to ensure that resources are distributed in a way that meets project requirements without exceeding available resources.

  3. Financial Planning: Inequalities are used in financial planning to model budget constraints and investment goals. For example, an individual might use inequalities to determine how much they can spend each month while still saving enough for retirement.

  4. Engineering: Engineers use inequalities in various applications, such as designing structures that can withstand certain loads or ensuring that systems operate within specified safety parameters.

  5. Economics: In economics, inequalities are used to model supply and demand, income distribution, and other economic phenomena. For instance, the Gini coefficient, a measure of income inequality, is based on the concept of inequalities.

Conclusion

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the steps involved and the rules that govern inequalities, such as reversing the inequality sign when multiplying or dividing by a negative number, you can accurately find solutions to a variety of problems. In the case of the inequality -8t + 2 > -75, we found that the solution is t < 9.625, and we verified that the given values t = 4, t = -2, t = -1, and t = -6 all satisfy this condition. This comprehensive exploration provides a solid foundation for tackling more complex inequalities and applying these skills in practical scenarios.

By avoiding common mistakes and practicing problem-solving techniques, you can enhance your ability to work with inequalities confidently. Whether you are a student learning algebra or a professional applying mathematical concepts in your field, a strong understanding of inequalities will undoubtedly be a valuable asset.

Final Answer

The solutions to the inequality -8t + 2 > -75 among the given options are t = 4, t = -2, t = -1, and t = -6.