Solving Inequalities A Step By Step Guide To (12x) ≤ -24

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Inequalities are a fundamental concept in mathematics, representing relationships where values are not necessarily equal. Understanding how to solve inequalities is crucial for various mathematical applications, from algebra to calculus. In this comprehensive guide, we will delve into the process of solving the linear inequality (12x) ≤ -24, providing a step-by-step explanation and illustrating the underlying principles.

Understanding Inequalities

Before we dive into the specifics of solving (12x) ≤ -24, let's first establish a solid understanding of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express a relationship of greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). These symbols define a range of values that satisfy the given condition.

In the context of (12x) ≤ -24, we are looking for all values of 'x' that, when multiplied by 12, result in a product less than or equal to -24. This means the solution will not be a single value, but rather a set of values within a specific interval.

Basic Properties of Inequalities

To effectively solve inequalities, we need to understand their fundamental properties. These properties govern how we can manipulate inequalities while preserving their validity. The key properties include:

  1. Addition/Subtraction Property: Adding or subtracting the same value from both sides of an inequality does not change the inequality's direction. For example, if a < b, then a + c < b + c and a - c < b - c.
  2. Multiplication/Division Property (Positive Value): Multiplying or dividing both sides of an inequality by the same positive value does not change the inequality's direction. For example, if a < b and c > 0, then ac < bc and a/c < b/c.
  3. Multiplication/Division Property (Negative Value): Multiplying or dividing both sides of an inequality by the same negative value reverses the inequality's direction. For example, if a < b and c < 0, then ac > bc and a/c > b/c.

This last property is crucial and often a point of error for students. Remember, when multiplying or dividing by a negative number, you must flip the inequality sign to maintain the correct relationship.

Solving Linear Inequalities: A Step-by-Step Approach

Now that we have a firm grasp of the basics, let's tackle the inequality (12x) ≤ -24. Solving linear inequalities involves isolating the variable 'x' on one side of the inequality. This is achieved by applying the properties of inequalities, much like solving equations.

Step 1: Isolate the Variable Term

Our goal is to get 'x' by itself on one side of the inequality. In the inequality (12x) ≤ -24, the variable term is 12x. To isolate this term, we need to undo the multiplication by 12. We can do this by dividing both sides of the inequality by 12.

(12x) / 12 ≤ -24 / 12

Step 2: Simplify the Inequality

Simplifying both sides of the inequality, we get:

x ≤ -2

This simplified inequality tells us that any value of 'x' less than or equal to -2 will satisfy the original inequality (12x) ≤ -24.

Step 3: Express the Solution Set

The solution to an inequality is not a single value, but a set of values. There are several ways to express this solution set:

  • Inequality Notation: This is the most direct way, simply stating the solution as x ≤ -2.
  • Interval Notation: Interval notation uses parentheses and brackets to represent the range of values. In this case, the solution is expressed as (-∞, -2]. The parenthesis indicates that -∞ is not included in the solution (as infinity is not a number), while the bracket indicates that -2 is included.
  • Graphical Representation: We can also represent the solution on a number line. Draw a number line, mark -2, and draw a closed circle (or a solid dot) at -2 to indicate that it is included in the solution. Then, shade the line to the left of -2, representing all values less than -2.

Common Mistakes to Avoid

When solving inequalities, it's important to be aware of common mistakes that students often make. Here are a few to keep in mind:

  • Forgetting to Flip the Inequality Sign: This is the most frequent error. Remember to flip the inequality sign whenever you multiply or divide both sides by a negative number.
  • Incorrectly Applying the Order of Operations: Just like with equations, follow the order of operations (PEMDAS/BODMAS) when simplifying inequalities.
  • Misinterpreting the Solution Set: Understand the difference between strict inequalities (<, >) and inclusive inequalities (≤, ≥), and how they are represented in interval notation and on a number line.
  • Not Checking the Solution: Always check your solution by plugging a value from your solution set back into the original inequality to ensure it holds true.

Example Applications

Solving inequalities is not just a theoretical exercise; it has practical applications in various fields. For example:

  • Budgeting: If you have a limited budget, you can use inequalities to determine how much you can spend on different items.
  • Physics: Inequalities can be used to describe ranges of physical quantities, such as temperature or velocity.
  • Optimization: In optimization problems, inequalities are used to define constraints on the variables.

Diving Deeper into Inequality Concepts

Building upon our understanding of solving the basic inequality (12x) ≤ -24, let's explore some advanced concepts related to inequalities. These concepts will broaden your mathematical toolkit and allow you to tackle more complex problems.

Compound Inequalities

Compound inequalities combine two or more inequalities using the words "and" or "or." These inequalities represent a more nuanced range of solutions.

  • "And" Inequalities: An "and" inequality requires that both inequalities be true simultaneously. The solution set is the intersection of the solution sets of the individual inequalities. For example, the compound inequality -3 < x ≤ 5 means that x must be greater than -3 and less than or equal to 5.
  • "Or" Inequalities: An "or" inequality requires that at least one of the inequalities be true. The solution set is the union of the solution sets of the individual inequalities. For example, the compound inequality x < -1 or x > 2 means that x must be less than -1 or greater than 2.

Solving compound inequalities involves solving each individual inequality separately and then combining the solution sets based on the "and" or "or" condition.

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars, such as |x - 2| < 3. The absolute value of a number represents its distance from zero, so these inequalities deal with distances.

To solve absolute value inequalities, we need to consider two cases:

  1. The expression inside the absolute value is positive or zero: In this case, we can simply remove the absolute value bars and solve the resulting inequality.
  2. The expression inside the absolute value is negative: In this case, we need to negate the expression inside the absolute value bars before removing them and solving the inequality. Remember to reverse the inequality sign when negating the expression.

For example, to solve |x - 2| < 3, we would consider the cases:

  • x - 2 < 3 (when x - 2 ≥ 0)
  • -(x - 2) < 3 (when x - 2 < 0)

Solving these two inequalities will give us the complete solution set.

Polynomial Inequalities

Polynomial inequalities involve polynomial expressions, such as x² - 3x + 2 > 0. Solving these inequalities requires a different approach than linear inequalities.

The general strategy for solving polynomial inequalities is as follows:

  1. Find the Roots: Set the polynomial expression equal to zero and solve for the roots. These roots are the points where the polynomial changes sign.
  2. Create a Sign Chart: Draw a number line and mark the roots on it. These roots divide the number line into intervals.
  3. Test Intervals: Choose a test value within each interval and plug it into the original polynomial inequality. Determine whether the polynomial is positive or negative in each interval.
  4. Identify the Solution Set: Based on the sign chart, identify the intervals that satisfy the inequality. Remember to consider whether the endpoints (roots) are included in the solution set based on the inequality symbol (>, <, ≥, ≤).

Systems of Inequalities

Just like we can have systems of equations, we can also have systems of inequalities. A system of inequalities consists of two or more inequalities involving the same variables. The solution to a system of inequalities is the set of all points that satisfy all the inequalities in the system.

The most common way to solve systems of inequalities is graphically. To do this:

  1. Graph Each Inequality: Graph each inequality on the same coordinate plane. Remember to use dashed lines for strict inequalities (>, <) and solid lines for inclusive inequalities (≥, ≤). Shade the region that satisfies each inequality.
  2. Identify the Feasible Region: The feasible region is the region where the shaded areas of all the inequalities overlap. This region represents the solution set to the system of inequalities.
  3. Find the Corner Points: The corner points of the feasible region are the points where the boundary lines intersect. These points are often important in optimization problems.

Applications of Advanced Inequality Concepts

The concepts of compound inequalities, absolute value inequalities, polynomial inequalities, and systems of inequalities have wide-ranging applications in various fields:

  • Optimization: Many optimization problems involve constraints that are expressed as inequalities. These inequalities define the feasible region within which the optimal solution must lie.
  • Calculus: Inequalities play a crucial role in calculus, particularly in determining intervals of increasing/decreasing functions, concavity, and limits.
  • Economics: Inequalities are used to model supply and demand, budget constraints, and other economic phenomena.
  • Computer Science: Inequalities are used in algorithms and data structures, such as sorting and searching algorithms.

By mastering these advanced inequality concepts, you will be well-equipped to tackle a wide range of mathematical and real-world problems.

Conclusion

Solving the inequality (12x) ≤ -24 is a fundamental exercise that lays the groundwork for understanding more complex mathematical concepts. By mastering the properties of inequalities and the step-by-step approach to solving them, you can confidently tackle a wide range of problems. Remember to pay close attention to the rules for multiplying and dividing by negative numbers, and always check your solution to ensure accuracy. This comprehensive guide has equipped you with the knowledge and skills necessary to confidently solve inequalities and apply them in various contexts. Keep practicing and exploring, and you will continue to expand your mathematical abilities.