Solving 3x + 15 ≤ 4x + 5 ≤ X + 13 Finding Integer Solutions
This article will walk you through the process of solving the compound inequality $3x + 15 \leq 4x + 5 \leq x + 13$. We will break down the inequality, solve each part separately, find the intersection of the solutions, and finally, identify the integer values of $x$ that satisfy the given conditions. Understanding how to solve compound inequalities is a fundamental skill in algebra, and this article aims to provide a clear and comprehensive explanation.
Understanding Compound Inequalities
Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or." In this case, we have a continuous compound inequality, which can be viewed as two inequalities connected by an "and." Specifically, $3x + 15 \leq 4x + 5 \leq x + 13$ is equivalent to the following two inequalities:
To solve the compound inequality, we need to find the values of $x$ that satisfy both inequalities simultaneously. This means we will solve each inequality individually and then find the intersection of their solution sets. This approach ensures that the final solution includes only the values of $x$ that meet all the conditions specified in the original compound inequality.
The first step in solving any inequality is to isolate the variable. This involves using algebraic manipulations, such as adding, subtracting, multiplying, or dividing both sides of the inequality, while maintaining the inequality's balance. Remember that multiplying or dividing by a negative number requires flipping the inequality sign. By systematically isolating the variable in each inequality, we can determine the range of values that satisfy each condition and ultimately find the solution to the compound inequality.
Solving the First Inequality: 3x + 15 ≤ 4x + 5
Let's start by solving the first inequality: $3x + 15 \leq 4x + 5$. Our goal is to isolate $x$ on one side of the inequality. To do this, we will follow these steps:
- Subtract $3x$ from both sides: This will move the $x$ term from the left side to the right side.
- Subtract 5 from both sides: This will isolate the $x$ term on the right side.
So, the solution to the first inequality is $10 \leq x$, which means that $x$ is greater than or equal to 10. This is a critical piece of the overall solution, as it sets a lower bound for the values of $x$ that satisfy the compound inequality. Any value of $x$ less than 10 will not satisfy the first part of the compound inequality, and therefore cannot be part of the final solution.
This result can also be written as $x \geq 10$, which is a more conventional way to express that $x$ is greater than or equal to 10. This form emphasizes that we are looking for values of $x$ that are at least 10. This lower bound will be crucial when we consider the solution to the second inequality and find the intersection of the two solution sets. The next step is to solve the second inequality in a similar manner, and then we can combine the results to find the overall solution.
Solving the Second Inequality: 4x + 5 ≤ x + 13
Now, let's solve the second inequality: $4x + 5 \leq x + 13$. Again, our aim is to isolate $x$ on one side of the inequality. Here are the steps we'll take:
- Subtract $x$ from both sides: This will move the $x$ term from the right side to the left side.
- Subtract 5 from both sides: This will isolate the term with $x$ on the left side.
- Divide both sides by 3: This will isolate $x$ on the left side.
Therefore, the solution to the second inequality is $x \leq \frac{8}{3}$. This means that $x$ is less than or equal to $\frac{8}{3}$, which is approximately 2.67. This provides an upper bound for the values of $x$ that satisfy the compound inequality. Any value of $x$ greater than $\frac{8}{3}$ will not satisfy the second part of the compound inequality and cannot be included in the final solution.
This upper bound, combined with the lower bound we found from the first inequality, will help us determine the specific range of values that satisfy the entire compound inequality. The next crucial step is to find the intersection of the solution sets from both inequalities, which will give us the final solution set for the compound inequality.
Finding the Intersection of the Solutions
To solve the compound inequality $3x + 15 \leq 4x + 5 \leq x + 13$, we need to find the values of $x$ that satisfy both inequalities we solved earlier:
The intersection of these two solution sets represents the values of $x$ that meet both conditions. In other words, we are looking for values of $x$ that are both greater than or equal to 10 and less than or equal to $\frac{8}{3}$.
However, we immediately notice a problem: 10 is a much larger number than $\frac{8}{3}$ (which is approximately 2.67). There are no numbers that can be both greater than or equal to 10 and less than or equal to 2.67. This means there is no overlap between the two solution sets.
In mathematical terms, the intersection of the two solution sets is an empty set, denoted by $\emptyset$. This indicates that there are no real numbers that satisfy both inequalities simultaneously. Therefore, the compound inequality $3x + 15 \leq 4x + 5 \leq x + 13$ has no solution in the set of real numbers.
This result is an important reminder that not all inequalities have solutions. Sometimes, the conditions imposed by the inequalities are contradictory, leading to an empty solution set. In this specific case, the requirement that $x$ be greater than or equal to 10 and simultaneously less than or equal to 2.67 creates a contradiction that cannot be satisfied. Understanding how to identify such contradictions is a key aspect of solving inequalities.
Identifying Integer Solutions
Since we have determined that there are no real number solutions to the compound inequality $3x + 15 \leq 4x + 5 \leq x + 13$, it follows that there are also no integer solutions. An integer is a whole number (not a fraction), which can be positive, negative, or zero. Because there are no real numbers that satisfy the compound inequality, there cannot be any integers that satisfy it either.
The solutions to the individual inequalities were:
-
x \geq 10$ (integers: 10, 11, 12, ...)
-
x \leq \frac{8}{3}$ (integers: ..., -1, 0, 1, 2)
As we discussed earlier, there is no overlap between these two sets of solutions. The integers greater than or equal to 10 are all larger than the integers less than or equal to $\frac{8}{3}$, so there are no integers that satisfy both conditions simultaneously.
Therefore, the set of integer solutions for the compound inequality is also an empty set, $\emptyset$. This result reinforces the importance of carefully analyzing the solutions of individual inequalities within a compound inequality to determine if there is a valid overall solution. In this case, the contradictory conditions lead to the absence of both real number and integer solutions.
Graphical Representation of the Solution
Visualizing the solution graphically can further clarify why there are no solutions to the compound inequality. We can represent the solutions to each inequality on a number line.
- For $x \geq 10$, we draw a closed circle at 10 (to indicate inclusion) and shade the number line to the right, representing all values greater than or equal to 10.
- For $x \leq \frac{8}{3}$, we draw a closed circle at $\frac{8}{3}$ (approximately 2.67) and shade the number line to the left, representing all values less than or equal to $\frac{8}{3}$.
When we look at these two shaded regions on the number line, we can see that they do not overlap. There is a gap between the values greater than or equal to 10 and the values less than or equal to $\frac{8}{3}$. This visual representation clearly demonstrates that there are no values of $x$ that satisfy both inequalities simultaneously.
The graphical representation serves as a powerful tool for understanding the nature of solutions to inequalities. It allows us to see the range of values that satisfy each individual inequality and to identify whether there is any overlap, which is necessary for a compound inequality to have a solution. In this case, the lack of overlap visually confirms our earlier conclusion that the compound inequality has no solution.
Conclusion
In summary, we solved the compound inequality $3x + 15 \leq 4x + 5 \leq x + 13$ by breaking it down into two separate inequalities, solving each one individually, and then finding the intersection of their solution sets. We found that the solution to the first inequality is $x \geq 10$, and the solution to the second inequality is $x \leq \frac{8}{3}$.
Because there are no values of $x$ that can simultaneously satisfy both conditions (being greater than or equal to 10 and less than or equal to $\frac{8}{3}$), the compound inequality has no solution. Consequently, there are also no integer values of $x$ that satisfy the inequality.
This problem illustrates an important concept in solving inequalities: not all inequalities have solutions. Sometimes, the conditions imposed by the inequalities are contradictory, resulting in an empty solution set. Understanding how to identify these contradictions is a crucial skill in algebra. By carefully analyzing each inequality and finding the intersection of their solutions, we can accurately determine whether a compound inequality has a solution and, if so, what that solution is.
Finally, the graphical representation further reinforces our conclusion by visually demonstrating the lack of overlap between the solution sets of the individual inequalities. This comprehensive approach ensures a thorough understanding of the problem and its solution.
