Solving Inequalities A Step-by-Step Guide To 2(2x + 3) - 10 < 6(x - 2)

In this comprehensive guide, we will delve into the process of solving inequalities, using the example 2(2x + 3) - 10 < 6(x - 2) as our primary focus. Inequalities, a fundamental concept in algebra, represent relationships where two expressions are not necessarily equal. Mastering the techniques to solve them is crucial for various mathematical and real-world applications. This guide aims to provide a clear, step-by-step approach to tackle inequalities, ensuring a solid understanding of the underlying principles and practical application.

Understanding Inequalities

Before diving into the specifics of solving the inequality 2(2x + 3) - 10 < 6(x - 2), it's essential to grasp the basics of inequalities. Inequalities are mathematical statements that compare two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which aim to find specific values that make the expressions equal, inequalities seek to find a range of values that satisfy the given condition. The solution to an inequality is often represented as an interval on the number line, indicating all the values that make the inequality true.

Key Concepts and Properties of Inequalities

1. Inequality Symbols: Understanding the meaning of each symbol is the first step. The symbol '<' denotes that the left-hand side is strictly less than the right-hand side. Conversely, '>' signifies that the left-hand side is strictly greater than the right-hand side. The symbols '≤' and '≥' include the possibility of equality; '≤' means less than or equal to, and '≥' means greater than or equal to.

2. Addition and Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the solution set. This property is a cornerstone of solving inequalities, allowing us to isolate the variable term on one side of the inequality.

3. Multiplication and Division Property: Multiplying or dividing both sides of an inequality by a positive number does not change the solution set. However, a crucial rule to remember is that multiplying or dividing by a negative number reverses the direction of the inequality sign. This is a critical distinction from solving equations and a common pitfall for students.

4. Graphical Representation: The solution to an inequality can be represented graphically on a number line. This visual representation helps in understanding the range of values that satisfy the inequality. Open circles are used to denote values that are not included in the solution (for strict inequalities like < and >), while closed circles indicate that the values are included (for inequalities with ≤ and ≥).

5. Interval Notation: Solutions to inequalities are often expressed using interval notation. This notation uses parentheses and brackets to indicate whether the endpoints are included in the solution. For example, (a, b) represents all numbers between a and b, excluding a and b. [a, b] represents all numbers between a and b, including a and b. Intervals can also be unbounded, using infinity symbols (∞ or -∞) to represent intervals that extend indefinitely in one or both directions.

By understanding these fundamental concepts, we lay a solid groundwork for tackling more complex inequalities. The properties of inequalities are the tools we'll use to manipulate and simplify the given expression, ultimately leading us to the solution.

Step-by-Step Solution of 2(2x + 3) - 10 < 6(x - 2)

Now, let's embark on the step-by-step solution of the inequality 2(2x + 3) - 10 < 6(x - 2). We will break down each step with clear explanations to ensure a thorough understanding of the process. This example demonstrates how the properties of inequalities are applied in practice to arrive at the solution.

Step 1: Distribute

The first step in solving this inequality involves distributing the numbers outside the parentheses to the terms inside. This process eliminates the parentheses and simplifies the inequality, making it easier to work with. Distribution is based on the distributive property of multiplication over addition and subtraction, which states that a(b + c) = ab + ac and a(b - c) = ab - ac.

Applying the distributive property to our inequality, we get:

2(2x + 3) - 10 < 6(x - 2)

(2 * 2x) + (2 * 3) - 10 < (6 * x) - (6 * 2)

This simplifies to:

4x + 6 - 10 < 6x - 12

Step 2: Combine Like Terms

After distributing, the next step is to combine any like terms on each side of the inequality. Like terms are terms that have the same variable raised to the same power. In this case, we can combine the constant terms on the left side of the inequality.

Looking at our inequality:

4x + 6 - 10 < 6x - 12

We can combine the constants 6 and -10 on the left side:

4x - 4 < 6x - 12

This simplification makes the inequality cleaner and sets the stage for isolating the variable.

Step 3: Isolate the Variable Term

The goal now is to isolate the variable term (the term with 'x') on one side of the inequality. To do this, we need to get all the 'x' terms on one side and all the constant terms on the other. A common strategy is to move the variable term with the smaller coefficient to the other side to avoid dealing with negative coefficients.

In our inequality:

4x - 4 < 6x - 12

We can subtract 4x from both sides to move the variable terms to the right side. This gives us:

4x - 4 - 4x < 6x - 12 - 4x

Simplifying, we get:

-4 < 2x - 12

Step 4: Isolate the Constant Term

Next, we want to isolate the constant term on the side opposite the variable. To do this, we add or subtract the constant term on the variable side from both sides of the inequality. In our case, we need to eliminate the -12 from the right side.

Starting with:

-4 < 2x - 12

We add 12 to both sides:

-4 + 12 < 2x - 12 + 12

This simplifies to:

8 < 2x

Step 5: Solve for the Variable

Finally, to solve for 'x', we need to isolate 'x' by dividing both sides of the inequality by the coefficient of 'x'. It's crucial to remember the rule about dividing by a negative number: if we divide or multiply by a negative number, we must reverse the inequality sign.

In our case, we have:

8 < 2x

We divide both sides by 2:

8 / 2 < 2x / 2

This gives us:

4 < x

This can also be written as:

x > 4

Step 6: Express the Solution

The solution to the inequality is x > 4. This means that any value of 'x' greater than 4 will satisfy the original inequality. It's important to express this solution in a clear and understandable manner. There are several ways to express the solution:.

1. Inequality Notation: The solution is already expressed in inequality notation: x > 4.
2. Graphical Representation: On a number line, we represent the solution by drawing an open circle at 4 and shading the line to the right, indicating all values greater than 4.
3. Interval Notation: In interval notation, the solution is represented as (4, ∞). The parenthesis indicates that 4 is not included in the solution, and ∞ represents infinity, indicating that the solution extends indefinitely to the right.

Verifying the Solution

After solving an inequality, it's a good practice to verify the solution to ensure accuracy. This can be done by selecting a value within the solution set and a value outside the solution set and plugging them into the original inequality. If the solution is correct, the value within the solution set should make the inequality true, while the value outside the solution set should make it false.

Test a Value Within the Solution Set

Let's choose x = 5, which is greater than 4 and therefore within our solution set. Substitute x = 5 into the original inequality:

2(2x + 3) - 10 < 6(x - 2)

2(2(5) + 3) - 10 < 6(5 - 2)

2(10 + 3) - 10 < 6(3)

2(13) - 10 < 18

26 - 10 < 18

16 < 18

This is true, so our solution holds for x = 5.

Test a Value Outside the Solution Set

Now, let's choose x = 4, which is not greater than 4 and therefore outside our solution set. Substitute x = 4 into the original inequality:

2(2x + 3) - 10 < 6(x - 2)

2(2(4) + 3) - 10 < 6(4 - 2)

2(8 + 3) - 10 < 6(2)

2(11) - 10 < 12

22 - 10 < 12

12 < 12

This is false (12 is not less than 12), confirming that x = 4 is not a solution. This verification process adds confidence to our solution and highlights the importance of checking our work.

Common Mistakes to Avoid

Solving inequalities, while similar to solving equations, has certain nuances that can lead to errors if not carefully addressed. Being aware of these common pitfalls can help in avoiding mistakes and ensuring accurate solutions. Here are some common mistakes to watch out for:

Forgetting to Reverse the Inequality Sign

The most common mistake in solving inequalities is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This is a critical step that stems from the fundamental properties of inequalities. When you multiply or divide by a negative number, you are essentially flipping the number line, which necessitates reversing the direction of the inequality.

For example, if you have -2x < 6, dividing both sides by -2 requires flipping the '<' sign to '>':

-2x / -2 > 6 / -2

x > -3

Incorrectly Distributing Negative Signs

Another common error occurs during the distribution step, especially when dealing with negative signs. It's essential to ensure that the negative sign is distributed correctly to all terms inside the parentheses. For instance, in the expression - (x - 3), the negative sign needs to be applied to both 'x' and '-3'.

-(x - 3) = -x + 3

Combining Like Terms Incorrectly

Combining like terms is a fundamental algebraic operation, but mistakes can happen if terms are not correctly identified or if arithmetic errors are made. Ensure that you are only combining terms that have the same variable raised to the same power. Additionally, double-check the arithmetic operations to avoid errors in addition or subtraction.

Misinterpreting Inequality Symbols

Misunderstanding the meaning of inequality symbols (>, <, ≥, ≤) can lead to incorrect solutions. Remember that '>' means greater than, '<' means less than, '≥' means greater than or equal to, and '≤' means less than or equal to. The inclusion or exclusion of the endpoint in the solution set depends on the symbol used.

Not Checking the Solution

Failing to verify the solution is a significant oversight. Checking the solution by plugging in values within and outside the solution set can reveal errors and provide confidence in the correctness of the answer. This step is crucial in catching mistakes and ensuring accuracy.

Conclusion

In conclusion, solving inequalities requires a systematic approach and a solid understanding of the underlying principles. By following the step-by-step guide outlined in this article, you can confidently tackle inequalities like 2(2x + 3) - 10 < 6(x - 2). Remember to distribute correctly, combine like terms carefully, isolate the variable term, and, most importantly, reverse the inequality sign when multiplying or dividing by a negative number. Always verify your solution to ensure accuracy. With practice and attention to detail, you can master the art of solving inequalities and apply this skill to various mathematical and real-world problems.