Solving 7/3x < 6: A Step-by-Step Guide
Hey everyone! Today, let's dive deep into solving the inequality (b) 7/3x < 6. Inequalities are a fundamental part of mathematics, and mastering them is crucial for various applications in real-world scenarios. In this guide, we'll break down the steps involved in solving this particular inequality, ensuring you grasp the underlying concepts and can tackle similar problems with confidence. We'll go through each step meticulously, providing clear explanations and insights. So, grab your calculators and notebooks, and let's get started!
Understanding Inequalities
Before we jump into the solution, let's quickly recap what inequalities are and how they differ from equations. While equations state that two expressions are equal, inequalities indicate a relationship where two expressions are not necessarily equal. Instead, they show that one expression is either greater than, less than, greater than or equal to, or less than or equal to another expression. These relationships are represented by the symbols >, <, ≥, and ≤, respectively.
In our case, we're dealing with the '<' symbol, which means 'less than'. This implies that the expression on the left side of the inequality (7/3x) is smaller than the expression on the right side (6). Our goal is to find all values of 'x' that satisfy this condition. Think of it like a range of possibilities, rather than a single solution.
To solve inequalities, we use many of the same techniques we use for solving equations, but there's one key difference to keep in mind: when we multiply or divide both sides of an inequality by a negative number, we need to flip the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the order of the numbers on the number line. For example, if 2 < 4, multiplying both sides by -1 gives -2 > -4. This rule is crucial for solving inequalities accurately.
Step-by-Step Solution of 7/3x < 6
Now, let's tackle the given inequality, 7/3x < 6, step by step. We'll break it down into manageable parts to ensure you understand each stage of the process.
Step 1: Isolate 'x'
Our primary goal is to isolate 'x' on one side of the inequality. Currently, 'x' is multiplied by 7/3. To get 'x' by itself, we need to undo this multiplication. The easiest way to do this is to multiply both sides of the inequality by the reciprocal of 7/3, which is 3/7. Remember, what we do to one side of the inequality, we must do to the other to maintain the balance.
So, we have:
(3/7) * (7/3)x < 6 * (3/7)
Multiplying the fractions on the left side, (3/7) * (7/3), simplifies to 1, leaving us with just 'x'. On the right side, we multiply 6 by 3/7.
x < 6 * (3/7)
Step 2: Simplify the Right Side
Next, we need to simplify the right side of the inequality. We have 6 multiplied by 3/7. To perform this multiplication, we can think of 6 as a fraction, 6/1. Then, we multiply the numerators (6 * 3) and the denominators (1 * 7).
x < (6/1) * (3/7) x < 18/7
So, we've simplified the right side to 18/7. This fraction represents the upper bound for 'x'.
Step 3: Interpret the Solution
Our solution is x < 18/7. This means that any value of 'x' that is less than 18/7 will satisfy the original inequality. To get a better sense of what this means, we can convert the improper fraction 18/7 to a mixed number. Dividing 18 by 7, we get 2 with a remainder of 4. So, 18/7 is equal to 2 4/7. Alternatively, we can convert 18/7 to a decimal, which is approximately 2.57.
Therefore, our solution can be interpreted as 'x' being less than 2 4/7 or approximately 2.57. This means that any number smaller than this value, such as 2, 1, 0, -1, and so on, will make the inequality 7/3x < 6 true. It's a range of solutions, not just a single value.
Representing the Solution
There are a couple of ways we can represent the solution x < 18/7.
1. Number Line
One visual way to represent the solution is using a number line. We draw a number line and mark the point 18/7 (or 2.57). Since 'x' is strictly less than 18/7, we use an open circle at this point to indicate that 18/7 is not included in the solution. Then, we shade the region to the left of 18/7, representing all the values less than 18/7. This shaded region shows all the possible values of 'x' that satisfy the inequality.
2. Interval Notation
Another way to represent the solution is using interval notation. Interval notation uses parentheses and brackets to indicate the range of values. In this case, since 'x' is less than 18/7 but does not include 18/7 itself, we use a parenthesis. The solution extends infinitely to the left, so we use negative infinity (-∞) as the lower bound. Infinity is always enclosed in a parenthesis because it's not a specific number but a concept of unboundedness.
So, the solution in interval notation is (-∞, 18/7). This notation clearly and concisely represents all the values of 'x' that satisfy the inequality.
Testing the Solution
To ensure our solution is correct, we can test it by plugging in values of 'x' that are within and outside the solution range into the original inequality. This helps us verify that our calculations are accurate and that we've correctly identified the solution set.
1. Choose a Value Within the Solution Range
Let's pick a value less than 18/7 (approximately 2.57). A simple choice is x = 2. Now, we substitute this value into the original inequality:
7/3x < 6 7/3 * 2 < 6 14/3 < 6
To compare 14/3 and 6, we can convert 6 to a fraction with a denominator of 3: 6 = 18/3.
14/3 < 18/3
This statement is true, so x = 2 is indeed a solution. This test supports our solution x < 18/7.
2. Choose a Value Outside the Solution Range
Now, let's pick a value greater than 18/7. A convenient choice is x = 3. Substitute this into the original inequality:
7/3x < 6 7/3 * 3 < 6 7 < 6
This statement is false. Therefore, x = 3 is not a solution, which aligns with our solution x < 18/7. This test further validates our solution.
3. Test the Boundary Point
Finally, let's test the boundary point, x = 18/7. We know that the inequality is strict (x < 18/7), so 18/7 should not be a solution. Substituting x = 18/7:
7/3x < 6 7/3 * (18/7) < 6 (7 * 18) / (3 * 7) < 6 18/3 < 6 6 < 6
This statement is false, as 6 is not less than 6. This confirms that 18/7 is not part of the solution set, which is consistent with our solution x < 18/7.
Common Mistakes to Avoid
When solving inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution.
1. Forgetting to Flip the Inequality Sign
As we mentioned earlier, the most crucial mistake to avoid is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. This mistake can lead to an incorrect solution set. Always double-check whether you've multiplied or divided by a negative number and, if so, remember to reverse the inequality sign.
2. Incorrectly Applying Operations
Another common mistake is performing incorrect arithmetic operations, such as adding, subtracting, multiplying, or dividing incorrectly. Ensure you're following the order of operations and that your calculations are accurate. A small error in arithmetic can lead to a completely different solution.
3. Misinterpreting the Inequality Sign
It's essential to correctly interpret the inequality signs. Make sure you understand the difference between < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). A misinterpretation can lead to representing the solution set incorrectly.
4. Not Testing the Solution
Failing to test the solution is another mistake. As we demonstrated, testing values within and outside the solution range helps you verify the accuracy of your solution. It's a simple yet effective way to catch errors.
Real-World Applications of Inequalities
Inequalities aren't just abstract mathematical concepts; they have numerous applications in the real world. Understanding and solving inequalities can help you make informed decisions in various situations.
1. Budgeting
Inequalities are commonly used in budgeting. For example, you might have a constraint on how much money you can spend each month. This can be expressed as an inequality, where your total expenses must be less than or equal to your budget. Solving this inequality helps you determine how much you can spend on different items while staying within your budget.
2. Speed Limits
Speed limits on roads are expressed as inequalities. The speed of a vehicle must be less than or equal to the posted speed limit. If 'v' represents the speed of the vehicle and 'L' is the speed limit, the inequality is v ≤ L. This ensures safety on the roads.
3. Manufacturing
In manufacturing, inequalities are used to set tolerances for product dimensions. For instance, the diameter of a bolt might need to be within a certain range. This can be expressed as an inequality, ensuring that the bolts meet quality standards.
4. Health and Fitness
Inequalities are also used in health and fitness. For example, a person's body mass index (BMI) should be within a certain range for a healthy weight. This range can be expressed as a compound inequality. Similarly, exercise recommendations often involve inequalities, such as exercising for at least a certain number of minutes per week.
Conclusion
Solving the inequality 7/3x < 6 is a fundamental exercise in algebra that illustrates the principles of solving inequalities. We've broken down the steps, from isolating 'x' to simplifying the solution and representing it on a number line and in interval notation. We also emphasized the importance of testing the solution and avoiding common mistakes.
Inequalities are a powerful tool in mathematics and have wide-ranging applications in everyday life. By mastering the techniques for solving inequalities, you'll be well-equipped to tackle a variety of problems and make informed decisions. So keep practicing, and you'll become an inequality-solving pro in no time! Remember, math is like a muscle; the more you exercise it, the stronger it gets. Keep up the great work, guys!
