Solving 4.2x + 5.6 < 7.2 - 8.3x A Step-by-Step Guide
Introduction
In this comprehensive guide, we will walk through the process of solving the inequality 4.2x + 5.6 < 7.2 - 8.3x step by step. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving an inequality means finding the range of values for the variable that makes the inequality true. This article aims to provide a clear, detailed explanation of each step involved in solving this particular inequality, making it easy for anyone to follow along and understand the underlying concepts.
Step 1: Isolating the Constant Term
The first crucial step in solving the inequality 4.2x + 5.6 < 7.2 - 8.3x involves isolating the constant term on one side of the inequality. In this case, we aim to move the constant terms to the right side of the inequality. The current inequality has two constant terms: 5.6 on the left side and 7.2 on the right side. To isolate the constant term, we need to eliminate 5.6 from the left side. We accomplish this by performing the inverse operation, which is subtraction. By subtracting 5.6 from both sides of the inequality, we maintain the balance and ensure that the inequality remains valid.
The process of subtracting 5.6 from both sides can be represented mathematically as follows:
4.2x + 5.6 - 5.6 < 7.2 - 8.3x - 5.6
This operation simplifies the inequality by removing the constant term from the left side, leaving us with the variable term and a new constant term on the right side. After performing the subtraction, the inequality becomes:
4.2x < 1.6 - 8.3x
This simplified form is much easier to work with as we move towards isolating the variable x. By isolating the constant term, we have taken a significant step towards solving the inequality. The next steps will involve gathering the variable terms on one side and then isolating the variable itself. Understanding this initial step is crucial as it sets the foundation for the subsequent algebraic manipulations required to find the solution.
Step 2: Gathering Variable Terms
Having isolated the constant term, the next strategic move in solving the inequality 4.2x < 1.6 - 8.3x is to gather all the variable terms on one side. This simplifies the inequality further and brings us closer to isolating the variable x. Currently, we have 4.2x on the left side and -8.3x on the right side. To consolidate these variable terms, we need to eliminate the -8.3x term from the right side. The inverse operation of subtraction is addition, so we will add 8.3x to both sides of the inequality. This ensures that we maintain the balance of the inequality while moving the variable term.
The mathematical representation of this step is:
4.2x + 8.3x < 1.6 - 8.3x + 8.3x
By adding 8.3x to both sides, we effectively cancel out the -8.3x term on the right side, leaving only the constant term. On the left side, we combine the 4.2x and 8.3x terms. This process results in the following simplified inequality:
12.5x < 1.6
Now, we have a single term involving x on the left side and a constant term on the right side. This form is much easier to work with, as the next step will involve isolating x by dividing both sides by the coefficient of x. Gathering variable terms is a crucial step in solving inequalities because it simplifies the equation and allows us to apply basic algebraic operations more effectively. This step sets the stage for the final isolation of the variable, which will reveal the solution set for the inequality.
Step 3: Isolating the Variable
With the inequality simplified to 12.5x < 1.6, the next critical step is to isolate the variable x. This involves removing the coefficient 12.5 from the left side of the inequality. The operation that connects 12.5 and x is multiplication, so to isolate x, we perform the inverse operation: division. We will divide both sides of the inequality by 12.5. It is important to remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be reversed. However, in this case, we are dividing by a positive number (12.5), so the inequality sign remains the same.
The division can be mathematically represented as follows:
(12.5x) / 12.5 < 1.6 / 12.5
Performing the division on both sides, we get:
x < 0.128
This result tells us that the solution to the inequality is all values of x that are less than 0.128. Isolating the variable is a fundamental step in solving any inequality or equation. It allows us to determine the specific range of values (in the case of inequalities) that satisfy the original statement. The solution x < 0.128 represents an infinite set of numbers, all of which are less than 0.128. This solution can be visualized on a number line, where all points to the left of 0.128 (not including 0.128) are part of the solution set.
Step 4: Expressing the Solution
Having isolated the variable and found the solution x < 0.128, the final step is to express this solution in a clear and understandable format. There are several ways to express the solution to an inequality, including using inequality notation, interval notation, and graphically on a number line. Each method offers a different way to represent the set of all values that satisfy the inequality 4.2x + 5.6 < 7.2 - 8.3x.
Inequality Notation:
The solution is already expressed in inequality notation: x < 0.128. This notation directly states that x can be any number less than 0.128. It’s a straightforward way to represent the solution set and is easy to understand.
Interval Notation:
Interval notation is another common way to express the solution set. It uses parentheses and brackets to indicate the range of values. For the inequality x < 0.128, the interval notation is (-∞, 0.128). The parenthesis next to -∞ and 0.128 indicates that these values are not included in the solution set. -∞ represents negative infinity, meaning the solution includes all numbers extending indefinitely in the negative direction. The value 0.128 is not included because the inequality is strictly “less than” and not “less than or equal to.”
Graphical Representation on a Number Line:
To represent the solution graphically, we draw a number line and mark the value 0.128 on it. Since x is less than 0.128, we draw an open circle at 0.128 (to indicate that it is not included in the solution) and shade the line to the left, representing all numbers less than 0.128. This visual representation provides a clear picture of the solution set.
In summary, the solution to the inequality 4.2x + 5.6 < 7.2 - 8.3x can be expressed as x < 0.128 in inequality notation, (-∞, 0.128) in interval notation, and as a shaded line on a number line extending from negative infinity up to (but not including) 0.128. Understanding these different methods of expressing solutions is crucial for effectively communicating mathematical results.
Conclusion
In this guide, we have meticulously walked through the steps required to solve the inequality 4.2x + 5.6 < 7.2 - 8.3x. Starting with the initial isolation of the constant term, we moved on to gathering variable terms, isolating the variable x, and finally expressing the solution in various notations. Each step is grounded in fundamental algebraic principles, ensuring a clear and logical progression towards the final solution: x < 0.128. This solution, expressed in inequality notation, tells us that any value of x less than 0.128 will satisfy the original inequality.
We also explored how to represent this solution using interval notation (-∞, 0.128), which provides a concise way to denote the range of values that satisfy the inequality. Additionally, we discussed the graphical representation of the solution on a number line, which offers a visual understanding of the solution set.
Understanding how to solve inequalities is a crucial skill in mathematics and has wide-ranging applications in various fields, including science, engineering, economics, and computer science. The ability to manipulate inequalities and find their solutions allows for the modeling and analysis of real-world problems involving constraints and limitations.
The techniques and steps outlined in this guide provide a solid foundation for tackling more complex inequalities and mathematical problems. By mastering these fundamental concepts, one can confidently approach a wide array of mathematical challenges.
