Math Problems: True Or False Statements & Conversions

Let's break down these math problems step by step, making sure everything is clear and easy to understand. We'll tackle the initial statement about adding centimeters to feet, and then dive into the true or false questions involving inequalities.

Is Adding 5 cm to a Foot Correct?

First, let's clarify the initial statement: "Adding 5 cm to a foot is correct." To determine if this statement is true or false, we need to understand what it means to "add" 5 cm to a foot in this context. It seems like we are checking some kind of calculation or comparison. We need to convert feet to centimeters or centimeters to feet to make a reasonable comparison.

• Conversion: 1 foot is approximately equal to 30.48 cm.
• Addition: Adding 5 cm to a foot (in centimeters) would mean 30.48 cm + 5 cm = 35.48 cm.
• Context: Without any context, it's challenging to determine the statement's truth. If the statement implies a specific result or equation, we need that information to evaluate. If we were to guess that the question is aimed at understanding unit conversion then the statement is vague. Let's consider the question is aiming to test the convertion of units.

So, is the statement true? It depends on what it is trying to say. If it's merely a calculation, then what calculation are we making? Let's explore what the possible answer is.

Understanding Unit Conversion and Precision

When dealing with unit conversions, especially in fields like engineering, physics, or even everyday measurements, precision and context are crucial. The conversion factor between feet and centimeters isn't just a fixed number; it represents a relationship that can be expressed with varying degrees of accuracy depending on the application.

For example, in construction, a difference of even a few millimeters can be significant when fitting materials together. In contrast, for estimating the height of a tree, rounding to the nearest centimeter might be perfectly acceptable. The context dictates the level of precision required.

The Role of Context in Evaluating Statements

Statements like "Adding 5 cm to a foot is correct" need context to be properly evaluated. Without context, the statement is vague and open to interpretation. It's similar to saying "Adding 2 and 2 is correct." While mathematically true in many cases, it might be incorrect if you're talking about combining two drops of a liquid, where surface tension might prevent them from merging into a single drop that's exactly twice the size. Understanding the scenario is paramount.

Now, let's dive into the real or false problems. Here are the solutions and explanations:

True or False: Inequalities

We are given that x is a real number such that x < 5. Let's evaluate each statement:

1. x - 7 ∈ [-2, +∞[

• Explanation: We need to check if x - 7 is greater than or equal to -2. Since x < 5, we have: x - 7 < 5 - 7 x - 7 < -2 This means x - 7 is less than -2, so it cannot be in the interval [-2, +∞[
• Answer: False

2. -2x ∈ [-10, +∞[

• Explanation: We need to check if -2x is greater than or equal to -10. Since x < 5, we multiply by -2 (and reverse the inequality): -2x > -2 * 5 -2x > -10 This means -2x is greater than -10, so it is in the interval [-10, +∞[
• Answer: True

3. 3x + 4 ∈ [-18, +∞[

• Explanation: We need to check if 3x + 4 is greater than or equal to -18. Since x < 5, we have: 3x < 3 * 5 3x < 15 3x + 4 < 15 + 4 3x + 4 < 19 Let's see if we can find a value of x < 5 such that 3x + 4 < -18 is false.

Since we want to know if 3x+4 is greater than -18, let's see if we can determine the range of this statement.

3x + 4 >= -18

3x >= -22

x >= -22/3 which is approximately -7.3.

So, we know x < 5, but x can be greater than -7.3. Therefore, 3x+4 can be greater than -18 and may fall in the given range.

• Answer: False

4. -x/2 + 1 >= 0

• Explanation: We need to check if -x/2 + 1 is greater than or equal to 0. Since x < 5, we have: -x > -5 -x/2 > -5/2 -x/2 + 1 > -5/2 + 1 -x/2 + 1 > -5/2 + 2/2 -x/2 + 1 > -3/2 Since -3/2 is -1.5, the expression -x/2 + 1 can be greater than 0, but it isn't always the case. Let's analyze it further.

To confirm the statement, we want to assess under what condition the expression -x/2 + 1 >= 0 holds true.

-x/2 + 1 >= 0

1 >= x/2

2 >= x

Since we are already given that x < 5, and we have determined that x must be less than or equal to 2, we know that this statement is not always true because x can be 3 or 4. 4 is less than 5, but would make the statement false.

• Answer: False

Final Answers

Here's a recap of the answers:

• Adding 5 cm to a foot is correct: Without context, we cannot answer.
• 1. x - 7 ∈ [-2, +∞[: False
• 2. -2x ∈ [-10, +∞[: True
• 3. 3x + 4 ∈ [-18, +∞[: False
• 4. -x/2 + 1 >= 0: False