Solving The Inequality 3(4-2t) ≥ 15 A Comprehensive Guide

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In this article, we will delve into the process of solving the inequality 3(4-2t) ≥ 15. Inequalities are mathematical statements that compare two expressions using symbols like ≥ (greater than or equal to), ≤ (less than or equal to), > (greater than), and < (less than). Solving an inequality means finding the range of values for the variable that make the inequality true. This particular inequality involves a variable t, and our goal is to isolate t on one side of the inequality to determine the solution set.

Understanding Inequalities

Before we jump into the solution, let's briefly discuss the fundamental concepts of inequalities. Inequalities are similar to equations, but instead of an equals sign (=), they use comparison symbols. When we solve an inequality, we are essentially finding all the values of the variable that satisfy the given condition. The solution to an inequality is often a range of values, rather than a single value as in the case of equations. This range can be represented graphically on a number line or in interval notation.

Key properties of inequalities that we'll use in solving this problem include:

  1. Addition/Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the inequality's direction.
  2. Multiplication/Division Property:
    • Multiplying or dividing both sides of an inequality by a positive number does not change the inequality's direction.
    • Multiplying or dividing both sides of an inequality by a negative number reverses the inequality's direction. This is a crucial rule to remember!

Step-by-Step Solution

Now, let's tackle the inequality 3(4-2t) ≥ 15 step by step:

Step 1: Distribute the 3

The first step in solving this inequality is to distribute the 3 across the terms inside the parentheses. This means multiplying 3 by both 4 and -2t:

3 * 4 - 3 * 2t ≥ 15

This simplifies to:

12 - 6t ≥ 15

Step 2: Isolate the Term with t

Our next goal is to isolate the term containing t (-6t) on one side of the inequality. To do this, we need to get rid of the 12 on the left side. We can accomplish this by subtracting 12 from both sides of the inequality. Remember, subtracting the same number from both sides doesn't change the direction of the inequality:

12 - 6t - 12 ≥ 15 - 12

This simplifies to:

-6t ≥ 3

Step 3: Solve for t

Now we have -6t greater than or equal to 3. To isolate t, we need to divide both sides of the inequality by -6. This is where the crucial rule about multiplying or dividing by a negative number comes into play: we must reverse the direction of the inequality sign:

(-6t) / (-6) ≤ 3 / (-6)

Notice that the ≥ symbol has changed to ≤. Simplifying the expression, we get:

t ≤ -1/2

Step 4: Express the Solution

The solution to the inequality is t ≤ -1/2. This means that any value of t that is less than or equal to -1/2 will satisfy the original inequality. We can express this solution in a few different ways:

  • Inequality Notation: t ≤ -1/2
  • Interval Notation: (-∞, -1/2]
  • Number Line: A number line with a closed circle (or bracket) at -1/2 and shading extending to the left, indicating all values less than or equal to -1/2.

Step 5: Verification

To verify the solution, we can pick a value of t that is less than or equal to -1/2 and substitute it back into the original inequality to see if it holds true. For example, let's choose t = -1:

3(4 - 2(-1)) ≥ 15 3(4 + 2) ≥ 15 3(6) ≥ 15 18 ≥ 15

This is true, so our solution is likely correct.

Common Mistakes and How to Avoid Them

Solving inequalities involves several steps where errors can occur. Here are some common mistakes and tips on how to avoid them:

  1. Forgetting to Distribute: Make sure to distribute any coefficients outside parentheses to all terms inside the parentheses. For example, in the first step of our problem, we needed to distribute the 3 to both the 4 and the -2t.
  2. Incorrectly Applying the Multiplication/Division Property: The most common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. Always double-check this step!
  3. Arithmetic Errors: Simple arithmetic mistakes can easily throw off the solution. Double-check your calculations at each step.
  4. Misinterpreting the Solution: Ensure you correctly interpret the solution set. For example, understand the difference between t < -1/2 and t ≤ -1/2. The latter includes -1/2 as a solution.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous applications in real-world scenarios. Here are a few examples:

  1. Budgeting: Inequalities can be used to represent budget constraints. For example, if you have a budget of $100 and want to buy two items, one costing $30 and the other costing x dollars, you can express this as an inequality: 30 + x ≤ 100.
  2. Temperature Ranges: Inequalities are used to define temperature ranges. For example, a certain chemical reaction might only occur within a temperature range of 20°C to 30°C, which can be expressed as 20 ≤ T ≤ 30, where T is the temperature.
  3. Speed Limits: Speed limits on roads are expressed as inequalities. For example, a speed limit of 65 mph can be written as v ≤ 65, where v is the vehicle's speed.
  4. Optimization Problems: Inequalities are used extensively in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints. These constraints are often expressed as inequalities.

Conclusion

Solving the inequality 3(4-2t) ≥ 15 involves a series of algebraic steps, including distribution, isolating the variable term, and dividing by a coefficient. The key takeaway is to remember to reverse the inequality sign when multiplying or dividing by a negative number. The solution, t ≤ -1/2, represents a range of values that satisfy the original inequality. Understanding how to solve inequalities is a crucial skill in mathematics with wide-ranging applications in various fields. By mastering these techniques and avoiding common mistakes, you can confidently tackle inequality problems.

We hope this comprehensive guide has been helpful in understanding how to solve the inequality 3(4-2t) ≥ 15. Remember to practice solving similar problems to reinforce your understanding and build your skills. Inequalities are a fundamental part of mathematics, and a solid grasp of them will be beneficial in many areas of study and in real-world situations.