Solving Inequalities A Comprehensive Guide With Examples

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In mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations, which assert the equality of two expressions, inequalities express the relative order of quantities. This comprehensive guide delves into the intricacies of solving inequalities, providing step-by-step explanations and practical examples to enhance your understanding. We will address two specific inequality problems, offering a detailed walkthrough of the solution process.

Understanding Inequalities

Before we dive into solving inequalities, let's establish a solid understanding of the fundamental concepts. An inequality is a mathematical statement that compares two expressions using inequality symbols. The most common inequality symbols are:

  • > : Greater than
  • < : Less than
  • ≥ : Greater than or equal to
  • ≤ : Less than or equal to

Inequalities are used extensively in various fields, including mathematics, physics, economics, and computer science, to model and solve real-world problems involving constraints, optimizations, and comparisons. Understanding how to manipulate and solve inequalities is therefore essential for students, researchers, and professionals alike.

The solutions to inequalities are often expressed as intervals on the number line, representing the range of values that satisfy the inequality. When solving inequalities, it is essential to remember certain rules that govern their manipulation, such as the effect of multiplying or dividing both sides by a negative number, which reverses the inequality sign. These rules ensure the solutions obtained are accurate and reflect the true range of possible values.

(i) Solving the Inequality: 3(2y + 1) ≥ 1/3(2y - 9)

Let's tackle our first inequality: 3(2y + 1) ≥ 1/3(2y - 9). This problem involves a linear inequality, which means the variable 'y' appears to the first power. To solve this, our goal is to isolate 'y' on one side of the inequality. Here’s a step-by-step breakdown:

Step 1: Distribute

Begin by distributing the constants on both sides of the inequality. This involves multiplying the numbers outside the parentheses by each term inside the parentheses. Distributing the constants is a fundamental step in simplifying the inequality and making it easier to work with.

  • On the left side, we have 3 multiplied by (2y + 1): 3 * 2y + 3 * 1 = 6y + 3
  • On the right side, we have 1/3 multiplied by (2y - 9): (1/3) * 2y - (1/3) * 9 = (2/3)y - 3

After distribution, our inequality looks like this: 6y + 3 ≥ (2/3)y - 3.

Step 2: Eliminate Fractions (Optional)

To simplify the inequality further, we can eliminate the fraction by multiplying every term by the least common multiple (LCM) of the denominators. In this case, the only denominator is 3, so we multiply every term by 3. Eliminating fractions makes the equation easier to manipulate and reduces the risk of errors in subsequent steps.

  • 3 * (6y + 3) ≥ 3 * ((2/3)y - 3)
  • 18y + 9 ≥ 2y - 9

Step 3: Gather 'y' Terms

Next, we want to group all the terms containing 'y' on one side of the inequality. To do this, subtract 2y from both sides of the inequality. Gathering like terms is a crucial step in solving any algebraic equation or inequality. This step helps in simplifying the expression and isolating the variable.

  • 18y - 2y + 9 ≥ 2y - 2y - 9
  • 16y + 9 ≥ -9

Step 4: Isolate the 'y' Term

Now, isolate the term with 'y' by subtracting 9 from both sides. This step involves performing the same operation on both sides of the inequality to maintain the balance. Isolating the variable term is essential for determining the range of values that satisfy the inequality.

  • 16y + 9 - 9 ≥ -9 - 9
  • 16y ≥ -18

Step 5: Solve for 'y'

Finally, solve for 'y' by dividing both sides of the inequality by 16. Since we are dividing by a positive number, the inequality sign remains the same. Dividing both sides by the coefficient of the variable is the final step in isolating the variable and finding the solution.

  • 16y / 16 ≥ -18 / 16
  • y ≥ -9/8

So, the solution to the inequality 3(2y + 1) ≥ 1/3(2y - 9) is y ≥ -9/8. This means that any value of 'y' that is greater than or equal to -9/8 will satisfy the original inequality.

Solution Set

The solution set can also be expressed in interval notation as [-9/8, ∞). This notation indicates that the solution includes all real numbers greater than or equal to -9/8. The square bracket indicates that -9/8 is included in the solution set, while the parenthesis indicates that infinity is not.

(ii) Solving the Inequality: 2/3 - 3x ≤ 2(1 - x)

Now, let's solve the second inequality: 2/3 - 3x ≤ 2(1 - x). This is another linear inequality, and we’ll follow a similar process to isolate 'x'. Solving this inequality will reinforce your understanding of the steps involved in solving inequalities and highlight some common techniques used in algebra.

Step 1: Distribute

First, distribute the constant on the right side of the inequality. Distributing constants is a fundamental algebraic operation that helps simplify expressions and solve equations or inequalities.

  • 2/3 - 3x ≤ 2 * 1 - 2 * x
  • 2/3 - 3x ≤ 2 - 2x

Step 2: Eliminate Fractions (Optional)

To eliminate the fraction, multiply every term in the inequality by 3. This simplifies the inequality by removing the fractional coefficient, making it easier to work with. Eliminating fractions is a common strategy in algebra to avoid computational errors and facilitate the solution process.

  • 3 * (2/3 - 3x) ≤ 3 * (2 - 2x)
  • 2 - 9x ≤ 6 - 6x

Step 3: Gather 'x' Terms

Next, gather all the terms containing 'x' on one side of the inequality. Add 9x to both sides of the inequality to group the x terms on the right side. Gathering like terms is a critical step in solving algebraic equations and inequalities, as it simplifies the expression and leads to the isolation of the variable.

  • 2 - 9x + 9x ≤ 6 - 6x + 9x
  • 2 ≤ 6 + 3x

Step 4: Isolate the 'x' Term

Isolate the term with 'x' by subtracting 6 from both sides of the inequality. Subtracting a constant from both sides is a valid algebraic manipulation that preserves the inequality and helps in isolating the variable term.

  • 2 - 6 ≤ 6 - 6 + 3x
  • -4 ≤ 3x

Step 5: Solve for 'x'

Finally, solve for 'x' by dividing both sides of the inequality by 3. Since we are dividing by a positive number, the inequality sign remains the same. Dividing by the coefficient of the variable is the final step in isolating the variable and finding the solution set.

  • -4 / 3 ≤ 3x / 3
  • -4/3 ≤ x

This can also be written as x ≥ -4/3. So, the solution to the inequality 2/3 - 3x ≤ 2(1 - x) is x ≥ -4/3. This means that any value of 'x' that is greater than or equal to -4/3 will satisfy the original inequality.

Solution Set

The solution set in interval notation is [-4/3, ∞). This interval includes all real numbers greater than or equal to -4/3. The square bracket indicates that -4/3 is included in the solution set, while the parenthesis indicates that infinity is not a bound.

Key Takeaways

Solving inequalities involves similar steps to solving equations, but with a crucial difference: multiplying or dividing by a negative number reverses the inequality sign. Understanding and applying this rule is essential for obtaining correct solutions. The process typically involves distributing constants, eliminating fractions, gathering like terms, isolating the variable term, and finally, solving for the variable. The solutions to inequalities are often expressed as intervals, representing a range of values that satisfy the inequality.

Conclusion

In this guide, we've explored how to solve inequalities, providing a detailed walkthrough of two specific examples. By understanding the fundamental concepts and practicing the steps involved, you can confidently tackle a wide range of inequality problems. Remember to pay close attention to the rules governing inequality manipulation and express your solutions clearly using interval notation when appropriate. Inequalities are a fundamental tool in mathematics and various applied fields, making the ability to solve them accurately an invaluable skill. Mastering these techniques will not only improve your mathematical proficiency but also enhance your problem-solving capabilities in real-world scenarios. Whether you are a student, educator, or professional, a solid understanding of inequalities is an asset that will serve you well in numerous contexts.