Problem 47 - Algebra and Trigonometry

Question

A phrase describing a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) at least 5 units from 7

Step-by-Step Solution

Answer

The inequality is \(|x - 7| \geq 5\).

Step 1: Understanding the Phrase

The phrase 'all real numbers \(x\) at least 5 units from 7' means that the distance between \(x\) and 7 is 5 or more. In terms of absolute value, this can be expressed by the inequality involving distance.

Step 2: Expressing Distance as an Inequality

To express the distance from \(x\) to 7, we use the absolute value \(|x - 7|\). The phrase 'at least 5 units' translates to \(|x - 7| \geq 5\) because the distance has to be 5 or more.

Step 3: Writing the Final Inequality

Combine the understanding from the previous steps to write the inequality: \(|x - 7| \geq 5\). This inequality represents all real numbers \(x\) that are at least 5 units away from 7.

Key Concepts

distance and absolute value

Understanding the concept of distance with absolute values is quite intriguing. It essentially addresses how far a number is from another on the number line, regardless of direction. In this context, an absolute value is just the positive version of a number or the distance from zero.

With absolute value inequalities, you can indicate the distance of a number from a specific point, like in the exercise "all real numbers \(x\) at least 5 units from 7." We represent this distance using the expression \(|x - 7|\). Here:

• The absolute value ensures that any negative distance is converted to positive.
• The inequality \(|x - 7| \geq 5\) shows numbers are at least 5 units away.

This is why absolute value is so crucial in portraying distances. It allows us to clearly and mathematically define a range of acceptable values.

real numbers

Real numbers are a foundational element in mathematics that include all the numbers you encounter on a number line. These numbers can be whole, decimal, positive, negative, and even irrational. Understanding real numbers is key to navigating through any math involving equations and inequalities.

The exercise mentions 'all real numbers \(x\).' It means that \(x\) can be any number on the number line, excluding imaginary numbers. With inequalities and absolute values, real numbers help express and solve real-world issues. These include:

• Distances and lengths, as demonstrated with the distance concept.
• Measurements in temperature, weight, or any continuous quantity.

Whenever you see 'real numbers' in a problem, it implies we are working within bounds that define all conceivable distances, without any jumps or gaps.

inequalities and equations

Inequalities and equations form the backbone of expressing mathematical relationships. While equations use the equal sign (\(=\)) to show that two values are the same, inequalities describe a range of possible solutions with signs like \(<\), \(>\), \(\leq\), and \(\geq\).

In our given solution, the inequality \(|x - 7| \geq 5\) indicates all numbers that are 5 or more units away from 7. This is crucial for understanding domains where specific ranges or conditions must be met without an exact value. Here are some essentials:

• Equations give exact solutions.
• Inequalities offer a range of solutions and conditions (like the distance in our exercise).
• They are crucial for optimization, boundaries, or dealing with minimum/maximum issues.

Understanding how inequalities and equations work together helps in solving many mathematical problems effectively, allowing for diverse solutions in both academic and real-world contexts.