Solving The Inequality -3(6-2x) ≥ 4x + 12 Finding The Interval Of Possible X Values
In this article, we delve into the mathematical inequality $-3(6-2x) ">=" 4x + 12$ to determine the interval that encompasses all possible values of $x$. This involves simplifying the inequality, isolating $x$, and expressing the solution set in interval notation. Understanding such inequalities is crucial in various mathematical contexts, including algebra, calculus, and real analysis. We will systematically break down the steps to solve this problem, providing a comprehensive explanation for each stage.
Understanding Inequalities
Before diving into the specifics of this inequality, it's essential to grasp the fundamental concepts 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 have definite solutions, inequalities often have a range of solutions. These solutions can be represented graphically on a number line or expressed in interval notation. When solving inequalities, certain operations, such as multiplying or dividing by a negative number, require flipping the inequality sign to maintain the correctness of the solution. The process of solving inequalities mirrors that of solving equations, but with the added consideration of the direction of the inequality.
Importance of Interval Notation
Interval notation is a way of writing subsets of the real number line. It is a concise and standardized method to represent the set of all numbers between two given endpoints. In interval notation, parentheses "()" are used to denote open intervals, which do not include the endpoints, while square brackets "[]" are used to denote closed intervals, which include the endpoints. For instance, the interval $(a, b)$ represents all real numbers between $a$ and $b$, excluding $a$ and $b$, whereas $[a, b]$ represents all real numbers between $a$ and $b$, including $a$ and $b$. The symbols $\infty$ (infinity) and $-\infty$ (negative infinity) are used to indicate intervals that extend indefinitely in the positive or negative direction, respectively. It is crucial to use parentheses with infinity symbols since infinity is not a specific number and cannot be included as an endpoint.
Solving Linear Inequalities
Solving linear inequalities involves a series of algebraic manipulations to isolate the variable on one side of the inequality. The goal is to find the range of values that satisfy the inequality. The steps typically include distributing terms, combining like terms, and performing operations on both sides of the inequality to isolate the variable. A critical rule to remember is that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For example, if we have the inequality $-x > 5$, multiplying both sides by -1 gives $x < -5$. This rule ensures that the solution set remains accurate.
Detailed Solution of the Inequality -3(6-2x) ≥ 4x + 12
To determine the interval that includes all possible values of $x$ for the inequality $-3(6-2x) ">=" 4x + 12$, we will follow a step-by-step approach, detailing each algebraic manipulation involved. Our goal is to isolate $x$ on one side of the inequality, thereby revealing the solution set. This process requires careful attention to the order of operations and the rules governing inequalities.
Step 1: Distribute the -3
The initial step involves distributing the $-3$ across the terms inside the parentheses. This simplifies the left side of the inequality and prepares it for further manipulation. Distributing $-3$ in $-3(6-2x)$ means multiplying both 6 and $-2x$ by $-3$. This gives us:
So, $-3(6-2x)$ becomes $-18 + 6x$. Rewriting the inequality, we have:
Step 2: Rearrange Terms to Isolate x
Next, we rearrange the terms to group the $x$ terms on one side of the inequality and the constants on the other side. This is done by adding or subtracting terms from both sides of the inequality. To isolate $x$, we will first subtract $4x$ from both sides:
This simplifies to:
Now, we add 18 to both sides of the inequality to isolate the term with $x$:
This simplifies to:
Step 3: Solve for x
To solve for $x$, we divide both sides of the inequality by 2. Since we are dividing by a positive number, we do not need to reverse the inequality sign:
This simplifies to:
Step 4: Express the Solution in Interval Notation
The solution $x \geq 15$ means that $x$ can be any value greater than or equal to 15. In interval notation, this is represented as $[15, \infty)$. The square bracket on 15 indicates that 15 is included in the solution set, and the infinity symbol indicates that the solution set extends indefinitely in the positive direction.
Selecting the Correct Option
Now that we have determined the solution to the inequality, we can match it with the given options. The solution $x \geq 15$ corresponds to the interval $[15, \infty)$.
The provided options were:
A. $(-\infty, -3]$ B. $[-3, \infty)$ C. $(-\infty, 15]$ D. $[15, \infty)$
Our solution $[15, \infty)$ matches option D.
Explanation of the Correct Answer
The correct option is D. $[15, \infty)$. This interval includes all values of $x$ that are greater than or equal to 15. The square bracket indicates that 15 is included in the solution, and the infinity symbol denotes that the interval extends without bound in the positive direction. This aligns perfectly with the solution we derived by solving the inequality $-3(6-2x) ">=" 4x + 12$.
Why Other Options Are Incorrect
Let's briefly discuss why the other options are incorrect:
- Option A. $(-\infty, -3]$: This interval represents all values of $x$ less than or equal to -3. This is not the solution we obtained, as our solution involves values greater than or equal to 15.
- Option B. $[-3, \infty)$: This interval includes all values of $x$ greater than or equal to -3. While it includes some positive values, it does not start at 15, making it an incorrect solution.
- Option C. $(-\infty, 15]$: This interval represents all values of $x$ less than or equal to 15. This is the opposite of our solution, which requires $x$ to be greater than or equal to 15.
Real-World Applications of Inequalities
Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including economics, engineering, and computer science, to model and solve problems involving constraints and optimization. Understanding how to solve inequalities is essential for making informed decisions in many practical situations.
Examples of Real-World Applications
- Budgeting and Finance: Inequalities can be used to determine how much money can be spent on different items while staying within a budget. For example, if a person has a budget of $100 and wants to buy two items, one costing $30, the inequality can help determine the maximum price of the second item.
- Engineering: In engineering, inequalities are used to set safety margins and ensure that structures can withstand certain loads. For instance, an engineer might use inequalities to calculate the maximum weight a bridge can support without exceeding its structural limits.
- Optimization Problems: Inequalities are fundamental in optimization problems, where the goal is to find the best solution within a set of constraints. These constraints are often expressed as inequalities. Linear programming, a technique widely used in operations research and management science, relies heavily on inequalities.
- Health and Medicine: Inequalities are used in medicine to define normal ranges for vital signs, such as blood pressure and cholesterol levels. They help healthcare professionals determine if a patient's measurements fall within a healthy range.
Importance in Mathematical Contexts
Inequalities are also crucial in various mathematical contexts, such as calculus and real analysis. They are used to define limits, continuity, and differentiability of functions. Understanding inequalities is essential for proving theorems and solving problems in these advanced mathematical fields.
Practice Problems
To reinforce your understanding of solving inequalities, let's work through a couple of practice problems.
Practice Problem 1
Solve the inequality $2(3x - 1) < 4x + 5$ and express the solution in interval notation.
Solution:
- Distribute the 2: $6x - 2 < 4x + 5$
- Subtract $4x$ from both sides: $2x - 2 < 5$
- Add 2 to both sides: $2x < 7$
- Divide by 2: $x < \frac{7}{2}$
- Express in interval notation: $(-\infty, \frac{7}{2})$
Practice Problem 2
Solve the inequality $-4(2x + 3) \geq -16$ and express the solution in interval notation.
Solution:
- Distribute the -4: $-8x - 12 \geq -16$
- Add 12 to both sides: $-8x \geq -4$
- Divide by -8 (and reverse the inequality sign): $x \leq \frac{1}{2}$
- Express in interval notation: $(-\infty, \frac{1}{2}]$
Conclusion
In summary, we have thoroughly explored the process of solving the inequality $-3(6-2x) ">=" 4x + 12$. By systematically applying algebraic manipulations, we determined that the solution is $x \geq 15$, which corresponds to the interval $[15, \infty)$. This detailed walkthrough highlights the importance of understanding inequalities and interval notation in mathematics. Additionally, we discussed the real-world applications of inequalities and provided practice problems to further enhance your understanding. Mastering these concepts is crucial for success in various mathematical and practical contexts.
Through this comprehensive analysis, we hope to have clarified the methods for solving inequalities and interpreting solutions in interval notation. The ability to solve inequalities is a fundamental skill in mathematics, with applications extending far beyond the classroom. By understanding the underlying principles and practicing regularly, you can confidently tackle a wide range of problems involving inequalities.
