Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use absolute value notation to define the interval (or pair of intervals) on the real number line. All real numbers more than 3 units from -1

Step-by-Step Solution

Answer

The intervals can be defined using absolute values as \( |x + 1| > 3 \). This represents all real numbers more than 3 units away from -1, with x < -4 or x > 2.

Step 1: Identifying intervals

Firstly, 'more than 3 units away from -1' means we have two intervals: from negative infinity to -4 and from 2 to positive infinity.

Step 2: Expressing intervals with absolute value notation

We express these two conditions in terms of absolute values. For all x less than -4, we have \( |-1 - x| > 3 \). For all x greater than 2, we have \( |x - (-1)| > 3 \). Simplifying, we get \( |x + 1| > 3 \) in both cases.

Step 3: Checking result

We have defined our intervals in terms of absolute values. To check if our answer is correct, we can select some values from these intervals and see if they satisfy the condition \( |x + 1| > 3 \). Let's check x=-5 (from the first interval) and x=3 (from the second interval). We have \( |-5 + 1| = 4 > 3 \) and \( |3 + 1| = 4 > 3 \), so our solution is correct.

Key Concepts

Real Number Line

The real number line is a visual representation of all real numbers, extending infinitely in both directions. Imagine it like a straight horizontal line where each point corresponds to a real number. You can think of it as a ruler with zero in the middle, negatives to the left, and positives to the right. Each number has a unique spot on this line.

• Zero acts as a central point.
• Negative numbers are found to the left of zero.
• Positive numbers lie to the right of zero.
• It can stretch infinitely in both directions.

When solving math problems, we often describe locations or ranges on this line to help us visualize mathematical concepts.

Intervals

Intervals are used to define a set of numbers that lie between two endpoints on the real number line. If you think of the real number line as a ruler, intervals are like specific segments on that ruler.

• Closed intervals include the endpoints and are denoted by square brackets, like \([a, b]\).
• Open intervals do not include the endpoints and use parentheses, like \( (a, b) \).
• There can also be a combination, like half-open intervals, where one end is included, and the other is not.

When expressing intervals, the endpoints dictate whether the interval is open or closed. Intervals help in defining parts of equations and inequalities, allowing us to solve problems by concentrating on specific segments of the number line.

Inequalities

Inequalities are mathematical expressions that indicate the relative size between two values. They help us see which number is greater, smaller, or if there's a relationship between them.

• The symbols used are '>' (greater than), '<' (less than), '≥' (greater than or equal to), and '≤' (less than or equal to).
• They are crucial when defining intervals.
• For example, when we say \(x > 5\), it indicates that x is any number greater than 5.

Understanding inequalities allows us to express a range of possible solutions or sets of data, especially when working with things like intervals or absolute values. They are the backbone of setting rules and conditions in many math problems.

Distance from a Point

Distance from a point on the real number line relates to how far a number is from another number, often the origin or another specified point. This concept is crucial when working with absolute values.

• Absolute value expressions like \(|x|\) show the distance of x from 0, ignoring direction.
• In the problem, the distance from -1 is expressed as \( |x + 1| > 3 \), meaning the numbers are more than 3 units away from -1.
• This can be visualized by imagining how far a point is from -1 in either direction on the number line.

When you grasp the idea of distance on the number line, problems involving movement, position, or separation become much easier to solve. By using absolute value notation, we can neatly express and solve these distance problems.