Problem 18 - Fundamentals of Mathematics

Question

Draw a number line that extends from -5 to 5 . Place points at all integers that satisfy \(-3 \leq x<4\).

Step-by-Step Solution

Answer

Mark points at -3, -2, -1, 0, 1, 2, and 3 on a number line from -5 to 5.

Step 1: Understanding the Number Line Requirement

We need to draw a number line from -5 to 5. This means we'll include all integers from -5 to 5 on the line.

Step 2: Identify the Range of Integers

The problem states the inequality \(-3 \leq x < 4\). This indicates that we need to find integers starting from -3 up to, but not including, 4.

Step 3: List the Integers that Satisfy the Inequality

The integers that satisfy this inequality are \(-3, -2, -1, 0, 1, 2, \text{ and } 3.\)

Step 4: Draw the Number Line

Draw a horizontal line, marking points at each integer value from -5 to 5.

Step 5: Place Points at the Specified Integers

On the number line, draw or mark points at -3, -2, -1, 0, 1, 2, and 3 because these are the integers that satisfy \(-3 \leq x < 4.\)

Key Concepts

Understanding Integers

Integers are whole numbers that do not have any fractional or decimal parts. They can be positive, negative, or zero. Basically, integers include numbers like -3, -2, -1, 0, 1, 2, and 3.
These numbers are found on the number line where each integer is evenly spaced from one another without any numbers in-between.
Remember, every integer has an opposite. For example, the opposite of 3 is -3. And every integer can be greater or less than another integer. This is key when dealing with numbers on a number line.

Reading Inequalities

Inequalities are used in mathematics to show that one number is less than or greater than another number. The symbol \(<\) means "less than," whereas \(>\) means "greater than."
For example, in the inequality \(-3 \leq x < 4\), the \(\leq\) symbol stands for "less than or equal to," indicating that \(x\) can be any number greater than or equal to -3.
The \(<\) symbol indicates that \(x\) can be any number less than 4. So, understanding inequalities helps us identify the specific numbers we need when working with a number line.

Identifying a Number Range

A number range describes a series of numbers between a set minimum and maximum value. For example, the range from -5 to 5 includes all integers from -5 through to and including 5.
It's crucial to be clear whether endpoints are included in this range. In the exercise, the range specified by \(-3 \leq x < 4\) includes -3, -2, -1, 0, 1, 2, and 3.
Knowing how to determine a number range helps in finding which integers to include when marking a number line.

Graphing Numbers on a Number Line

Graphing numbers requires plotting points corresponding to integers on a number line. This helps visually represent inequalities and number ranges.
To graph numbers between -3 and less than 4, draw and label a number line from -5 to 5. Then, place dots on the integers -3 through 3, which satisfy the inequality \(-3 \leq x < 4\).
By marking these integers on the line, it becomes easy to see which numbers are included in a particular inequality or range.