Graphing & Solving: (35x-14) ≤ 21x/2 + 3 Inequality
Let's dive into how to graph the solution of the inequality and determine its solution set. Understanding inequalities is crucial in mathematics, as they allow us to define ranges and boundaries rather than just specific values. This article breaks down the process step-by-step, making it easy to follow along, whether you're a student tackling homework or just refreshing your math skills. We’ll cover everything from simplifying the inequality to representing the solution graphically and in interval notation.
Understanding Inequalities
Before we jump into the specifics, let's recap what inequalities are all about. Inequalities are mathematical expressions that use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to compare values. Unlike equations, which show exact equivalences, inequalities show a range of possible values.
In our case, we have . The “≤” symbol means “less than or equal to,” so we’re looking for all values of x that make the left side of the expression less than or equal to the right side. To effectively work with inequalities, it’s essential to grasp the core principles that guide their manipulation and solutions. This foundational understanding sets the stage for tackling more complex problems and real-world applications, where constraints and ranges are often more pertinent than precise figures.
When solving inequalities, we aim to isolate the variable on one side, much like solving equations. However, there's a crucial rule to remember: if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This ensures that the relationship between the expressions remains accurate. For example, if we have -2x < 4, dividing both sides by -2 gives us x > -2 (note how the '<' sign flipped to '>'). The solution to an inequality isn't just a single number, but a set of numbers. This set can be represented graphically on a number line, showing the range of values that satisfy the inequality. We use open circles (o) to indicate values that are not included in the solution set (for inequalities with < or >) and closed circles (●) for values that are included (for inequalities with ≤ or ≥). Furthermore, the solution set can be expressed in interval notation, which provides a concise way to represent a range of numbers. For instance, x > 3 can be written as (3, ∞), indicating all numbers greater than 3, not including 3 itself. Conversely, x ≤ 5 is expressed as (-∞, 5], which includes all numbers less than or equal to 5. Understanding these nuances is key to accurately solving and interpreting inequalities.
Step-by-Step Solution
Now, let's break down the solution to our inequality into manageable steps.
1. Simplify the Inequality
Our first goal is to get rid of the fraction. To do this, we'll multiply both sides of the inequality by 2:
This simplifies to:
Simplifying an inequality is akin to clearing the underbrush in a dense forest; it reveals the underlying structure and makes the path forward clearer. This initial step of multiplying both sides by 2 serves to eliminate the fraction, a common hurdle in algebraic manipulations. By doing so, we transform the inequality into a more manageable form, making subsequent steps like combining like terms and isolating the variable significantly easier.
This process is not just about mathematical neatness; it’s about enhancing clarity and reducing the likelihood of errors. Fractions, while fundamental, can often introduce complexities that distract from the core solving process. By removing them, we focus more directly on the relationship between the variable and the constants, thus promoting a more intuitive understanding of the inequality. This foundational step is essential, as it streamlines the entire solution, setting the stage for the logical progression of algebraic operations that follow. Furthermore, it reinforces the basic principle of algebraic manipulation – maintaining balance. Multiplying both sides by the same number ensures that the inequality's truth remains unchanged, a crucial concept in solving any algebraic problem.
2. Isolate the Variable
Next, we want to get all the x terms on one side and the constants on the other. Let’s subtract 21x from both sides:
Which gives us:
Now, add 28 to both sides:
Simplifying further:
Isolating the variable in an inequality is akin to pinpointing the central figure in a complex puzzle. This process involves strategically maneuvering terms across the inequality to ensure that the variable stands alone, revealing its range of possible values. The steps of subtracting 21x from both sides and adding 28 to both sides are not arbitrary; they are deliberate actions that chip away at the complexity, gradually unveiling the solution. This careful manipulation is grounded in the fundamental principle of maintaining balance, ensuring that any operation performed on one side is mirrored on the other, thus preserving the integrity of the inequality.
The goal is not merely to rearrange terms but to distill the essence of the inequality, clarifying the relationship between the variable and the constants. Each step taken, from the initial subtraction to the final addition, brings us closer to understanding the boundaries within which x can exist. This isolation process is a cornerstone of algebraic problem-solving, extending beyond inequalities to equations and more complex mathematical constructs.
3. Solve for x
To find the solution set, we need to divide both sides by 49:
This results in:
So, the solution set includes all values of x that are less than or equal to 34/49.
Solving for x is like finding the last piece of a jigsaw puzzle, the final step that completes the picture. Dividing both sides of the inequality by 49 is a decisive action that isolates x, revealing the precise boundary of its solution set. This step is crucial because it translates the abstract inequality into a concrete statement about the possible values of x. The result, x ≤ 34/49, is not just a number; it’s a declaration that x can be any value as long as it doesn't exceed 34/49. This understanding is key to interpreting the inequality and applying it in various contexts.
The process of dividing to solve for x is also a testament to the elegance of algebraic manipulation. It demonstrates how a complex inequality can be systematically reduced to a simple, understandable form. Each step, from simplifying the initial expression to this final division, is a logical progression that builds upon the previous one, culminating in a clear and concise solution. This clarity is invaluable, especially in fields where precision and accuracy are paramount. Moreover, this step highlights the importance of understanding the properties of inequalities, ensuring that the division is carried out correctly and the direction of the inequality is maintained. The solution x ≤ 34/49 is not just an answer; it’s a gateway to further exploration, such as graphing the solution set or applying it to real-world scenarios.
Graphing the Solution
Now that we have the solution set (), let's graph it on a number line.
1. Draw a Number Line
Start by drawing a horizontal line. Mark zero somewhere in the middle and then add some values to the left and right to give it scale.
2. Locate 34/49
The value 34/49 is less than 1 but greater than 0. Mark this point on the number line. Since the inequality includes “equal to” (≤), we’ll use a closed circle (●) to indicate that 34/49 is included in the solution.
3. Shade the Solution Set
Since x is less than or equal to 34/49, we’ll shade the region to the left of our marked point. This shaded region represents all the values of x that satisfy the inequality.
Graphing the solution to an inequality transforms an abstract algebraic concept into a visual representation, making it more intuitive and accessible. Drawing a number line is the first step in this process, providing a canvas upon which the solution set can be depicted. The number line, with its infinite span in both directions, symbolizes the continuum of real numbers, and our task is to highlight the portion that satisfies the inequality. Locating 34/49 on this line is like pinpointing a specific address on a map. This value serves as the boundary of our solution set, and its accurate placement is crucial for a correct graph. The choice of using a closed circle (●) to mark 34/49 is deliberate and significant. It signifies that this value is included in the solution, a direct reflection of the “less than or equal to” (≤) symbol in the inequality.
The closed circle is not just a visual marker; it’s a mathematical statement. Shading the region to the left of 34/49 is the final stroke in our graphical representation. This shaded area represents the infinite set of numbers that are less than 34/49, all of which satisfy our inequality. The shading is a powerful visual tool, instantly conveying the scope of the solution and making it clear that x can take on any value within this range. This graphical representation is more than just a picture; it’s a synthesis of algebraic and geometric thinking, bridging the gap between symbols and space. It provides a holistic understanding of the solution, one that resonates beyond the confines of numerical values.
Expressing the Solution Set
We can express the solution set in a couple of ways:
1. Inequality Notation
As we found, the solution is simply:
2. Interval Notation
In interval notation, we use parentheses and brackets to represent intervals of numbers. Parentheses indicate that the endpoint is not included, while brackets indicate that it is. Since our solution includes all values less than or equal to 34/49, we use a bracket on the right and negative infinity on the left:
Expressing the solution set of an inequality in different notations is like translating a message into various languages, each offering a unique perspective on the same core information. Inequality notation, represented by x ≤ 34/49, is the most direct translation, mirroring the algebraic solution. It succinctly states the condition that x must satisfy, providing a clear and unambiguous definition of the solution set. However, inequality notation is just one way to convey this information. Interval notation offers a more compact and visually oriented representation. It uses parentheses and brackets to define the range of values, with parentheses indicating exclusion and brackets indicating inclusion of the endpoint.
In our case, (-∞, 34/49] encapsulates the same solution set as x ≤ 34/49, but it does so in a way that emphasizes the interval nature of the solution. The use of -∞ signifies that the solution extends indefinitely in the negative direction, while the bracket at 34/49 denotes that this value is included in the set. This notation is particularly useful in advanced mathematical contexts, where dealing with intervals and sets is commonplace. Understanding both inequality and interval notations is crucial for a comprehensive grasp of mathematical expressions. Each notation offers a distinct advantage, and the ability to fluently switch between them enhances one’s mathematical literacy. It’s like being bilingual in the language of mathematics, capable of communicating the same ideas in multiple forms.
Conclusion
Solving and graphing inequalities might seem daunting at first, but by breaking it down into steps, it becomes quite manageable. Remember to simplify, isolate the variable, and then express your solution in both inequality and interval notation. And don't forget to graph it on a number line for a clear visual representation!
For further exploration of inequalities and their applications, you might find resources like Khan Academy helpful.
