Solving Inequalities If X+12 ≤ 5-y And 5-y ≤ 2(x-3)

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In the realm of mathematics, inequalities play a pivotal role in defining the relationships between quantities. This article delves into a specific inequality problem, providing a comprehensive solution and a step-by-step explanation to enhance understanding. We will dissect the problem: If $x+12 \leq 5-y$ and $5-y \leq 2(x-3)$, then which statement is true?

Understanding the Problem

The problem at hand presents us with two inequalities: $x+12 \leq 5-y$ and $5-y \leq 2(x-3)$. The objective is to determine which of the given statements logically follows from these inequalities. This requires a careful analysis of the relationships between $x$ and $y$ as defined by these inequalities.

Breaking Down the Inequalities

Let's first examine each inequality separately:

1. $x+12 \leq 5-y$: This inequality states that the sum of $x$ and 12 is less than or equal to the difference between 5 and $y$. It establishes a connection between $x$ and $y$, indicating that as $x$ increases, $5-y$ must also increase or remain constant to maintain the inequality.
2. $5-y \leq 2(x-3)$: This inequality states that the difference between 5 and $y$ is less than or equal to twice the difference between $x$ and 3. This further refines the relationship between $x$ and $y$, adding another layer of constraint.

The Transitive Property of Inequalities

A crucial concept to grasp here is the transitive property of inequalities. This property states that if $a \leq b$ and $b \leq c$, then it logically follows that $a \leq c$. This principle will be instrumental in connecting the two given inequalities.

Solving the Problem

Now, let's apply the transitive property to our problem. We have:

• $x+12 \leq 5-y$ (Inequality 1)
• $5-y \leq 2(x-3)$ (Inequality 2)

According to the transitive property, since $x+12$ is less than or equal to $5-y$, and $5-y$ is less than or equal to $2(x-3)$, then $x+12$ must also be less than or equal to $2(x-3)$.

Identifying the Correct Statement

Therefore, the statement that holds true is:

C. $x+12 \leq 2(x-3)$

This statement directly follows from the transitive property applied to the given inequalities. The other options can be disproven by exploring different values of x and y that satisfy the initial inequalities.

Why Other Options Are Incorrect

To solidify our understanding, let's briefly examine why the other options are not correct:

• A. $x+12 \leq 2(5-y)$: This statement is not a direct consequence of the transitive property. While it might hold true for specific values of $x$ and $y$, it's not a guaranteed outcome based on the given inequalities.
• B. $x+12 \leq 2x-3$: This statement can be obtained by simplifying option C, $x+12 \leq 2(x-3)$, which expands to $x+12 \leq 2x-6$. However, the original option B is $x+12 \leq 2x-3$, which is a different inequality and may not always hold true given the initial conditions.
• D. $x+12 \leq y-5$: This statement contradicts the given inequalities. If $x+12$ is less than or equal to $5-y$, it cannot simultaneously be less than or equal to $y-5$ for all possible values of $x$ and $y$ that satisfy the initial conditions.

Detailed Explanation of Option C: $x+12 \leq 2(x-3)$

To further emphasize the correctness of option C, let's delve into a more detailed explanation. We've established that this statement is a direct result of the transitive property. However, let's break down the implications of this inequality and how it aligns with the original inequalities.

Expanding the Inequality

First, let's expand the right side of the inequality:

$x+12 \leq 2(x-3)$ becomes $x+12 \leq 2x-6$

Isolating x

Now, let's rearrange the inequality to isolate $x$:

Subtract $x$ from both sides: $12 \leq x-6$

Add 6 to both sides: $18 \leq x$

This resulting inequality, $18 \leq x$, tells us that $x$ must be greater than or equal to 18. This is a crucial piece of information that helps us understand the possible values of $x$ and their relationship with $y$.

Connecting to the Original Inequalities

Now, let's see how this constraint on $x$ aligns with the original inequalities.

1. $x+12 \leq 5-y$: If $x$ is greater than or equal to 18, then $x+12$ will be greater than or equal to 30. For this inequality to hold, $5-y$ must also be greater than or equal to 30. This implies that $y$ must be less than or equal to -25.
2. $5-y \leq 2(x-3)$: Let's substitute $x \geq 18$ into this inequality. The right side becomes $2(x-3) \geq 2(18-3) = 30$. For this inequality to hold, $5-y$ must be less than or equal to 30, which implies that $y$ must be greater than or equal to -25.

Consistency and Validity

Notice the consistency here. From the first inequality, we deduced that $y \leq -25$, and from the second inequality, we found that $y \geq -25$. The only value of $y$ that satisfies both conditions is $y = -25$. This consistency strengthens the validity of our conclusion that $x$ must be greater than or equal to 18 for the original inequalities to hold.

Verifying with Example Values

To further illustrate this, let's take an example. If we choose $x = 18$ and $y = -25$, the original inequalities become:

1. $18+12 \leq 5-(-25)$, which simplifies to $30 \leq 30$ (True)
2. $5-(-25) \leq 2(18-3)$, which simplifies to $30 \leq 30$ (True)

Both inequalities hold true. Now, let's check option C: $18+12 \leq 2(18-3)$, which simplifies to $30 \leq 30$ (True). This confirms that option C holds true for these values of $x$ and $y$.

The Importance of Transitive Property

This detailed explanation underscores the importance of the transitive property in solving inequality problems. By recognizing this property and applying it correctly, we can efficiently deduce the correct statement from a set of given inequalities.

Common Mistakes and How to Avoid Them

When dealing with inequalities, several common mistakes can lead to incorrect solutions. Understanding these pitfalls and learning how to avoid them is crucial for mastering inequality problems.

Mistake 1: Incorrectly Applying the Transitive Property

The transitive property states that if $a \leq b$ and $b \leq c$, then $a \leq c$. A common mistake is to misapply this property or to assume it applies in situations where it doesn't. For instance, one might incorrectly conclude that if $a < b$ and $b > c$, then $a < c$, which is not necessarily true.

How to Avoid It: Always double-check the direction of the inequality signs before applying the transitive property. Ensure that the inequalities are chained correctly, with the same direction of inequality between consecutive terms.

Mistake 2: Ignoring the Impact of Negative Signs

Multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign. This is a fundamental rule that is often overlooked, leading to incorrect conclusions.

How to Avoid It: Be extra cautious when multiplying or dividing inequalities by negative numbers. Remember to flip the inequality sign to maintain the correctness of the statement. For example, if $-x < 5$, then multiplying both sides by -1 gives $x > -5$.

Mistake 3: Making Assumptions About Variable Signs

Another common mistake is to make unwarranted assumptions about the signs of variables. For example, assuming that $x$ is positive when it could be negative or zero can lead to incorrect simplifications or conclusions.

How to Avoid It: Avoid making assumptions about the signs of variables unless explicitly stated in the problem. Consider all possible cases (positive, negative, and zero) to ensure a comprehensive solution.

Mistake 4: Incorrectly Combining Inequalities

When dealing with multiple inequalities, it's crucial to combine them correctly. Adding or subtracting inequalities requires careful attention to the direction of the inequality signs. For instance, if $a < b$ and $c < d$, then $a+c < b+d$. However, there are no general rules for subtracting or dividing inequalities directly.

How to Avoid It: When combining inequalities, pay close attention to the operations being performed. Addition of inequalities with the same direction is generally safe, but subtraction or division requires more careful consideration.

Mistake 5: Not Testing the Solution

After solving an inequality problem, it's always a good practice to test the solution by plugging in some values. This helps to verify the correctness of the solution and identify any potential errors.

How to Avoid It: Choose a few values within the solution range and substitute them into the original inequalities. If the inequalities hold true, the solution is likely correct. If not, there's a mistake that needs to be identified and corrected.

Conclusion

In summary, the correct answer to the problem is C. $x+12 \leq 2(x-3)$. This conclusion is reached by applying the transitive property of inequalities to the given statements. A thorough understanding of inequality properties, along with careful attention to detail, is essential for solving these types of problems accurately. By avoiding common mistakes and practicing regularly, one can develop proficiency in tackling inequality challenges.

This exploration of the inequality problem demonstrates the importance of understanding fundamental mathematical principles and applying them logically. By breaking down the problem, utilizing the transitive property, and considering potential pitfalls, we can confidently arrive at the correct solution and enhance our problem-solving skills.

Keywords

• Inequalities
• Transitive property
• Mathematical problem-solving
• Algebra
• Inequality solution
• Solving for x and y
• Common mistakes in inequalities
• Algebraic expressions
• Mathematical reasoning
• Inequality analysis