Question

Give a verbal description of the subset of real numbers that is represented by the inequality, and sketch the subset on the real number line.\(x \geq 4\)

Short answer

The inequality \(x \geq 4\) represents the set of all real numbers that are either 4 or greater than 4. On number line, this is displayed as all the points to the right of, and including, 4.

Step-by-step solution

Interpret the inequality

The inequality \(x \geq 4\) signifies that 'x' should be any number that is greater than or equal to 4. This includes 4 itself as well as numbers like 5, 6, 7, and even decimal values such as 4.5, 4.7, 5.3 etc. In fact, any real number that lies to the right of 4 on the number line is included.

Describe the subset verbally

The subset that is represented by this inequality is the set of all real numbers that are either 4 or greater than 4. This set includes positive integers greater than or equal to 4, all positive decimal numbers greater than 4, and all positive fractions where the numerator is greater than the denominator times 4.

Sketch the subset

The subset is drawn on the real number line by shading the portion of the line that is greater than or includes 4. This is done by making a solid dot at 4 (to show inclusion of 4 in the set) and then shading or drawing an arrow pointing towards the right (to represent all the numbers greater than 4).

Key concepts

Understanding Real Numbers

Real numbers include all the numbers we typically use in everyday life. They encompass:

• Whole numbers: like 0, 1, 2, etc.
• Integers: which include negative numbers like -3 and positive numbers like 5.
• Rational numbers: numbers that can be expressed as fractions, such as \( \frac{1}{2} \) or \( 0.75 \).
• Irrational numbers: numbers that cannot be expressed as exact fractions, such as \( \pi \) or the square root of a non-perfect square.

When we deal with inequalities like \( x \geq 4 \), we're referring to real numbers. These include not just integers, but also decimals and fractions. The versatility of real numbers allows us to describe any range or set, whether it's finite or infinite, on the number line.

Number Line Representation

A number line is a visual tool used to represent real numbers. When sketching the inequality \( x \geq 4 \):

• Start by marking the number 4 on the line.
• Make a solid dot at 4 to show that 4 is included in the set. A solid dot indicates that the number is part of the solution.
• Draw an arrow or shade the line to the right of 4. This represents all numbers greater than 4.

This visualization helps in understanding which numbers satisfy the inequality. It’s a helpful way to quickly see the part of the number line we're interested in. For \( x \geq 4 \), the line extends infinitely to the right, capturing all the numbers from 4 onwards.

Creating a Verbal Description of Sets

Verbal descriptions help us communicate the nature of a set without relying on mathematical symbols. For \( x \geq 4 \), a verbal description might be:

• "The set of all real numbers that are greater than or equal to 4."
• It includes numbers such as 4, 4.1, 5, and even non-integers like 4.5.
• Since these are all real numbers, the set is not limited to whole numbers, but extends to decimals and fractions as well.

Describing sets this way ensures that anyone, even those unfamiliar with the math symbols, can understand which numbers are included. It’s a bridge between mathematical notation and everyday language.