Problem 13 - Elementary Algebra

Question

Draw a number line that extends from -5 to 3 and place points at all numbers greater than or equal to -4 but strictly less than 2 .

Step-by-Step Solution

Answer

Answer: The numbers that satisfy this condition are -4, -3, -2, -1, 0, and 1.

Step 1: Draw the number line

Draw a horizontal line and mark the numbers -5 to 3 on it, including both ends. This will be the number line to work on.

Step 2: Identify the numbers in the given condition

We need to identify the numbers which are greater than or equal to -4 but strictly less than 2. This can be written in interval notation as [-4, 2). The numbers that satisfy this condition are -4, -3, -2, -1, 0, and 1.

Step 3: Place points on the number line

Place points above the number line at the numbers identified in Step 2 (-4, -3, -2, -1, 0, and 1).

Step 4: Label the range on the number line

Indicate the range of numbers greater than or equal to -4 but strictly less than 2 by placing a bracket around the series of points starting at -4 and ending at 2. The bracket should be closed at -4 (square bracket), and open at 2 (round parenthesis) to show that -4 is included in the range, but 2 is not.

Key Concepts

Interval Notation

Interval notation is a shorthand used in mathematics to describe a range of numbers along a number line. It's particularly useful when dealing with inequalities and algebra. To visualize this concept, let's think of a number line as a long street, where each house represents a number.

Now, imagine if we only want to consider certain houses on our street – maybe we’re only interested at those between house number -4 and 2. In interval notation, we write this as \[ -4, 2) \], where the square bracket indicates that -4 is the first house we include (in mathematical terms, -4 is 'included' in our interval), and the round parenthesis tells us that house number 2 is not included (in mathematical terms, 2 is 'not included' in our interval). This is a powerful way to compactly express a range of values, ensuring that anyone who reads it understands exactly which numbers are part of the set we're interested in.

Inequalities

Inequalities are like the rules that tell us how one number relates to another. They are the stops signs and speed limits of mathematics, indicating numbers that are larger or smaller, and whether they are allowed to be equal to a specific value. The inequality \(x \geq -4\) translates to 'x is greater than or equal to -4,' which includes -4 and all numbers to the right on the number line.

On the other end, we have the inequality \(x < 2\), which means 'x is strictly less than 2,' excluding the number 2 itself and considering only the numbers to its left. When combined, these inequalities form a range: \(-4 \leq x < 2\), implying that x can be any number starting from -4 up to but not including 2. Understanding inequalities is essential to correctly plot ranges on a number line and solve algebraic problems.

Plotting Points

Plotting points is similar to mapping out a treasure hunt; each number has a specific place to be found on the number line. To spot these numerical treasures, we begin by drawing the number line, which typically runs horizontally. Marks are evenly spaced to represent each integer, much like a ruler.

Upon receiving our map - which, in this case, is the interval [-4, 2) - we start placing marks on the number line at -4, -3, -2, -1, 0, and 1. These are the points where our treasure is buried. The square bracket at -4 tells us that's exactly where our treasure hunt begins – we mark it boldly to celebrate its importance. The parenthesis at 2 is the point where we stop; it's like a 'do not dig beyond this point' sign. This way, anyone who follows our map, or number line, will know precisely which points are part of our hunt and which are not.