Solving Inequalities Finding The Range Of X For 5 + X ≤ 8 And 13 + X ≥ 7
Understanding inequalities and how to solve them is a fundamental concept in mathematics, crucial for various applications across different fields. In this article, we will delve into the process of finding the range of values for x that satisfy two given inequalities: 5 + x ≤ 8 and 13 + x ≥ 7. This exercise will not only enhance your understanding of inequality operations but also demonstrate how to combine solutions from multiple inequalities to determine a common range.
Solving the First Inequality: 5 + x ≤ 8
To begin, let's tackle the first inequality, 5 + x ≤ 8. Our primary goal here is to isolate x on one side of the inequality. This can be achieved by performing operations that maintain the inequality's balance. The key operation we'll use is subtraction. We need to eliminate the 5 from the left side of the inequality, and we can do this by subtracting 5 from both sides. It is crucial to remember that whatever operation is performed on one side of an inequality must also be performed on the other side to preserve the relationship. This principle ensures that the solutions we find are accurate and valid.
Subtracting 5 from both sides of the inequality 5 + x ≤ 8 gives us:
5 + x - 5 ≤ 8 - 5
This simplifies to:
x ≤ 3
This result, x ≤ 3, is a significant milestone. It tells us that any value of x that is less than or equal to 3 will satisfy the first inequality. This includes values like 3, 2, 1, 0, -1, and so on, extending infinitely in the negative direction. Understanding this range is crucial for the next steps, where we will consider the second inequality and how its solutions overlap with this one. The ability to correctly solve and interpret such inequalities is a foundational skill in algebra and is essential for tackling more complex mathematical problems. The solution x ≤ 3 represents a half-line on the number line, starting from 3 and extending to negative infinity. This visual representation can be very helpful in understanding the set of numbers that satisfy the inequality.
Solving the Second Inequality: 13 + x ≥ 7
Now, let's move on to the second inequality, 13 + x ≥ 7. Similar to the first inequality, our objective is to isolate x on one side to determine the values that satisfy the condition. The process involves using inverse operations to undo the addition of 13. In this case, we will subtract 13 from both sides of the inequality. This ensures that we maintain the balance and the relationship between the two sides, a critical principle in solving inequalities.
Subtracting 13 from both sides of 13 + x ≥ 7, we get:
13 + x - 13 ≥ 7 - 13
Simplifying this, we find:
x ≥ -6
This result, x ≥ -6, provides us with another range of values for x. It indicates that any value of x that is greater than or equal to -6 will satisfy the second inequality. This includes -6, -5, -4, and extends indefinitely in the positive direction. Understanding this range is just as important as understanding the range from the first inequality. The next step is to combine these two ranges to find the common values that satisfy both inequalities simultaneously. The solution x ≥ -6 also represents a half-line on the number line, but this time it starts from -6 and extends towards positive infinity. The overlap between this half-line and the one representing x ≤ 3 will give us the final solution to the problem.
Combining the Inequalities to Find the Common Range
Having solved both inequalities individually, the next crucial step is to combine the solutions to find the range of values for x that satisfy both 5 + x ≤ 8 and 13 + x ≥ 7. This involves identifying the overlap between the solution sets of the two inequalities. The solution to the first inequality, 5 + x ≤ 8, is x ≤ 3. This means x can be any value less than or equal to 3. The solution to the second inequality, 13 + x ≥ 7, is x ≥ -6. This indicates that x must be greater than or equal to -6. To find the common range, we need to determine the values of x that meet both conditions simultaneously. In other words, we're looking for the values of x that are both less than or equal to 3 and greater than or equal to -6. This can be visualized on a number line, where one range extends from negative infinity up to 3, and the other range extends from -6 to positive infinity. The overlapping segment represents the solution set for the combined inequalities.
The common range for x is therefore -6 ≤ x ≤ 3. This means that x can be any number between -6 and 3, inclusive. To clarify, this includes -6, 3, and all numbers in between, such as -5, -4, -3, -2, -1, 0, 1, and 2. The endpoints -6 and 3 are included because the inequalities include the “equal to” condition (≤ and ≥). This type of combined inequality is known as a compound inequality, and it's a common concept in algebra. The solution -6 ≤ x ≤ 3 is a closed interval, meaning it includes its endpoints. This is a direct consequence of the inequalities being non-strict (i.e., including the “equal to” part). Understanding how to combine inequalities like this is essential for solving more complex problems, especially in calculus and real analysis, where intervals and ranges play a central role.
Representing the Solution on a Number Line
A number line provides a visual representation of the solution to the combined inequalities, -6 ≤ x ≤ 3, making it easier to understand the range of values that satisfy both conditions. To represent this solution graphically, we first draw a number line, which is a straight line representing all real numbers. We then mark the key points of our solution range, which are -6 and 3. Since the inequalities include the “equal to” condition (≤ and ≥), we use closed circles or brackets at -6 and 3 to indicate that these values are included in the solution set. If the inequalities were strict (i.e., < or >), we would use open circles or parentheses to show that the endpoints are not included.
Next, we shade the region on the number line between -6 and 3. This shaded segment represents all the numbers that fall within the range -6 ≤ x ≤ 3. Any point within this shaded region represents a valid solution to the compound inequality. Points outside this region do not satisfy both inequalities simultaneously. The number line representation offers an immediate visual confirmation of the solution. It clearly shows the boundaries of the solution set and the continuous range of values that x can take. This visual aid is particularly useful when dealing with more complex inequalities or when combining multiple inequalities, as it helps to prevent errors and ensures a clear understanding of the solution.
Importance of Understanding Inequalities in Mathematics
Understanding inequalities is fundamental in mathematics and has broad applications across various branches, including algebra, calculus, and real analysis. Inequalities are used to express relationships where quantities are not necessarily equal but have a specific order or range. They play a crucial role in problem-solving, especially in situations where exact values are not as important as understanding the boundaries or constraints within which a solution must lie. In algebra, inequalities are used to solve a wide range of problems, from simple linear inequalities to more complex polynomial and rational inequalities. They are essential for determining the feasible region in linear programming, a technique used to optimize solutions in various fields, such as economics and engineering. The ability to manipulate and solve inequalities is a cornerstone of algebraic proficiency. In calculus, inequalities are essential for defining intervals, analyzing function behavior, and determining limits and continuity. For instance, the epsilon-delta definition of a limit relies heavily on inequalities. Understanding the behavior of functions within specific intervals is critical for many applications of calculus, including optimization and finding areas and volumes.
In real analysis, inequalities form the basis for defining convergence, continuity, and other fundamental concepts. Many theorems in real analysis, such as the squeeze theorem and the Cauchy-Schwarz inequality, are based on inequalities. A solid understanding of inequalities is necessary for comprehending the rigorous proofs and concepts in real analysis. Beyond pure mathematics, inequalities are used extensively in various applied fields. In economics, they are used to model budget constraints, supply and demand relationships, and optimization problems. In physics, inequalities are used to describe physical constraints and to analyze the behavior of systems under certain conditions. In computer science, they are used in algorithm analysis, complexity theory, and optimization problems. The ability to work with inequalities is a valuable skill in many quantitative fields. Mastering inequalities involves understanding the basic operations that can be performed on them while preserving their validity, such as adding or subtracting the same quantity from both sides, multiplying or dividing by a positive number, and the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number. It also involves the ability to combine multiple inequalities, represent solutions graphically on a number line, and interpret the results in the context of the problem. In summary, a strong foundation in inequalities is indispensable for success in mathematics and many related disciplines.
Conclusion
In conclusion, finding the range of values for x that satisfy the inequalities 5 + x ≤ 8 and 13 + x ≥ 7 involves solving each inequality separately and then identifying the common solution set. By isolating x in each inequality, we found that x ≤ 3 and x ≥ -6. Combining these solutions, we determined that the range of values for x that satisfy both inequalities is -6 ≤ x ≤ 3. This range includes all numbers between -6 and 3, inclusive. The visual representation on a number line further clarifies this solution, showing a shaded segment between -6 and 3, with closed circles at the endpoints to indicate their inclusion. This exercise highlights the importance of understanding and manipulating inequalities, a crucial skill in mathematics with broad applications across various fields. Inequalities are not just mathematical tools; they are fundamental concepts that allow us to describe and analyze relationships in a wide range of contexts, from simple everyday situations to complex scientific and engineering problems. Mastering inequalities provides a solid foundation for further study in mathematics and related disciplines, enabling one to tackle more advanced topics and real-world problems with confidence. The ability to solve inequalities and interpret their solutions is an essential skill for anyone pursuing a quantitative field.
