Solving Inequalities A Step-by-Step Guide To 3t + 9 ≥ 15
In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries. Understanding how to solve inequalities is fundamental for various applications, from simple problem-solving to complex mathematical modeling. This article delves into the process of solving the inequality 3t + 9 ≥ 15, providing a clear and concise step-by-step guide suitable for students and anyone seeking to enhance their mathematical skills. Let's embark on this mathematical journey and unravel the solution together.
Understanding Inequalities
Before we dive into the specifics of solving 3t + 9 ≥ 15, it's essential to grasp the core concept of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another. The symbols >, <, ≥, and ≤ are the key indicators of these relationships. An inequality, like an equation, involves variables, constants, and mathematical operations, but instead of finding a single value that satisfies the expression, we seek a range of values that make the inequality true.
Inequalities are more than just abstract mathematical constructs; they have practical applications in various real-world scenarios. For instance, they can be used to define budget constraints, where spending must be less than or equal to a certain amount. In physics, inequalities can describe the range of possible values for a physical quantity, such as the speed of an object. Understanding inequalities, therefore, is not only crucial for mathematical proficiency but also for problem-solving in diverse fields.
In the context of solving inequalities, the goal is to isolate the variable on one side of the inequality sign. This process involves applying algebraic operations similar to those used in solving equations, with a crucial difference: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This rule is paramount and requires careful attention to ensure accurate solutions. The solution to an inequality is typically expressed as a range of values, which can be represented graphically on a number line, providing a visual representation of the possible solutions.
Step-by-Step Solution of 3t + 9 ≥ 15
Now, let's tackle the inequality 3t + 9 ≥ 15 step-by-step. Our aim is to isolate the variable 't' on one side of the inequality. This process mirrors solving equations, but with the added consideration of the inequality sign. Each step we take must maintain the balance of the inequality, ensuring that the relationship between the two sides remains valid.
Step 1: Isolate the Term with the Variable
The first step in solving the inequality 3t + 9 ≥ 15 is to isolate the term containing the variable 't'. This involves eliminating the constant term on the same side of the inequality. In this case, we need to remove the '+ 9'. To do this, we apply the inverse operation, which is subtraction. We subtract 9 from both sides of the inequality. This ensures that the inequality remains balanced, as we are performing the same operation on both sides. The result of this step is:
3t + 9 - 9 ≥ 15 - 9
Simplifying this, we get:
3t ≥ 6
This step brings us closer to isolating 't'. The constant term is now removed from the left side, leaving only the term with the variable. This simplification is a crucial step in the process of solving for 't'.
Step 2: Isolate the Variable
The next step in solving the inequality 3t ≥ 6 is to isolate the variable 't' completely. Currently, 't' is being multiplied by 3. To isolate 't', we need to undo this multiplication. The inverse operation of multiplication is division, so we will divide both sides of the inequality by 3. It's important to remember that when dividing (or multiplying) an inequality by a negative number, we must reverse the direction of the inequality sign. However, in this case, we are dividing by a positive number (3), so we do not need to reverse the sign.
Dividing both sides by 3, we get:
(3t) / 3 ≥ 6 / 3
Simplifying this, we get:
t ≥ 2
This is the solution to the inequality. It tells us that 't' is greater than or equal to 2. This means that any value of 't' that is 2 or greater will satisfy the original inequality 3t + 9 ≥ 15. We have now successfully isolated the variable and found the range of values that make the inequality true.
Solution and its Interpretation: t ≥ 2
The solution to the inequality 3t + 9 ≥ 15 is t ≥ 2. This concise statement encapsulates a range of possible values for 't'. It signifies that any value of 't' that is greater than or equal to 2 will satisfy the original inequality. This is a crucial distinction from equations, where the solution is typically a single value. Inequalities, on the other hand, provide a range of solutions, reflecting the inherent nature of relationships that are not strictly equal.
Understanding the Solution Set
The solution t ≥ 2 represents an infinite set of values. This set includes 2 itself, as the inequality includes the 'equal to' condition. It also encompasses all real numbers greater than 2, extending indefinitely towards positive infinity. This can be visualized on a number line, where a closed circle (or a square bracket) is placed at 2, and a line extends to the right, indicating all values greater than 2. This graphical representation provides a clear and intuitive understanding of the solution set.
Verification of the Solution
To ensure the correctness of the solution, it's essential to verify it. This can be done by substituting values from the solution set back into the original inequality. For instance, let's substitute t = 2 into the inequality 3t + 9 ≥ 15:
3(2) + 9 ≥ 15
6 + 9 ≥ 15
15 ≥ 15
This is true, confirming that t = 2 is indeed a solution. Now, let's try a value greater than 2, such as t = 3:
3(3) + 9 ≥ 15
9 + 9 ≥ 15
18 ≥ 15
This is also true, further validating our solution. Conversely, if we were to substitute a value less than 2, such as t = 1, we would find that the inequality is not satisfied:
3(1) + 9 ≥ 15
3 + 9 ≥ 15
12 ≥ 15
This is false, demonstrating that values less than 2 are not part of the solution set. This verification process solidifies our confidence in the accuracy of the solution.
Real-World Implications
The solution t ≥ 2 is not just an abstract mathematical result; it can have practical implications in various real-world scenarios. Imagine a situation where 't' represents the number of hours you need to work to earn at least $15, given that you earn $3 per hour and have already earned $9. The inequality 3t + 9 ≥ 15 models this scenario, and the solution t ≥ 2 indicates that you need to work at least 2 hours to meet your earning goal. This simple example illustrates how inequalities and their solutions can be used to model and solve everyday problems.
In other contexts, 't' could represent the minimum score needed on a test to achieve a certain grade, the minimum number of items to sell to reach a sales target, or the minimum amount of time needed to complete a task. The versatility of inequalities makes them a valuable tool for decision-making and problem-solving in a wide range of fields.
Conclusion
In conclusion, solving the inequality 3t + 9 ≥ 15 involves a systematic approach of isolating the variable 't'. By applying inverse operations and maintaining the balance of the inequality, we arrive at the solution t ≥ 2. This solution represents a range of values, all numbers greater than or equal to 2, that satisfy the original inequality. Understanding inequalities and their solutions is a fundamental skill in mathematics, with applications extending far beyond the classroom. From modeling real-world scenarios to making informed decisions, the ability to solve inequalities is a valuable asset in both academic and practical pursuits.
