Question

On a number line indicate all numbers greater than or equal to \(-\frac{1}{2}\) but less than \(\frac{3}{4}\).

Short answer

Question: Represent all numbers greater than or equal to -1/2 and less than 3/4 on a number line. Answer: To represent the given interval on the number line, draw a horizontal line and mark the points -1/2 and 3/4. Indicate that -1/2 is included in the interval with a solid circle or square bracket ([), and show that 3/4 is not included in the interval with an open circle or round bracket ( ). Then, draw a line segment between these two points, including the point at -1/2 and excluding the point at 3/4. This line segment represents all numbers greater than or equal to -1/2 but less than 3/4 on the number line.

Step-by-step solution

Draw a number line

First, draw a horizontal line. This line represents the number line, where each point on the line corresponds to a number, and the distance between points represents the difference between the numbers.

Mark the endpoints

On the number line, locate the points that correspond to \(-\frac{1}{2}\) and \(\frac{3}{4}\). These are the endpoints of the interval representing the numbers greater than or equal to \(-\frac{1}{2}\) and less than \(\frac{3}{4}\). To make these points more visible, draw a small vertical line segment at these points.

Indicate the interval

Now, we need to show the interval on the number line. Since the numbers in the interval are greater than or equal to \(-\frac{1}{2}\), draw a solid circle or square bracket ( [ ) at \(-\frac{1}{2}\) to indicate that this point is included in the interval. Then, since the numbers in the interval are less than \(\frac{3}{4}\), draw an open circle or round bracket ( ) at \(\frac{3}{4}\) to indicate that this point is not included in the interval.

Draw a line connecting the two points

To represent the numbers in the interval, draw a line segment between the points \(-\frac{1}{2}\) and \(\frac{3}{4}\) on the number line, including the point at \(-\frac{1}{2}\) and excluding the point at \(\frac{3}{4}\). This line segment represents all the numbers greater than or equal to \(-\frac{1}{2}\) but less than \(\frac{3}{4}\) on the number line. Now, the number line visually represents all numbers greater than or equal to \(-\frac{1}{2}\) but less than \(\frac{3}{4}\).

Key concepts

Interval Notation

Interval notation is a way to describe a set of numbers on a number line using brackets and numbers to indicate which parts of the line are included in the interval. It provides a concise and clear method to represent continuous ranges of values. For an interval that includes numbers greater than or equal to a certain value (let's say \(-\frac{1}{2}\)) and less than another value (e.g., \(\frac{3}{4}\)), interval notation is written with square brackets \([\) for numbers that are included and round parentheses \(()\) for numbers that are excluded.

In our exercise, the interval notation for all numbers greater than or equal to \(-\frac{1}{2}\) but less than \(\frac{3}{4}\) would be written as \([-\frac{1}{2}, \frac{3}{4})\). Here are key points about interval notation to keep in mind:

• Square brackets \([\text{ or } ]\) indicate that the end number is included in the interval.
• Round brackets \((\text{ or } )\) indicate that the end number is not included in the interval.
• They clearly map the start and end points of an interval lightly, providing an efficient way to express ranges of numbers.

Inequalities

Inequalities are mathematical statements that describe a range of possible solutions or values based on certain conditions. They use symbols like greater than \(>\), less than \(<\), greater than or equal to \(\geq\), and less than or equal to \(\leq\). Understanding these symbols is crucial as they tell us which numbers fall inside or outside the specified range.

In the context of our exercise, the inequality \(-\frac{1}{2} \leq x < \frac{3}{4}\) succinctly expresses the range of values for \(x\) that satisfy the specified condition:

• The symbol \( \leq \) informs us that \(-\frac{1}{2}\) is part of the solution, meaning \(x\) can equal \(-\frac{1}{2}\).
• The symbol \(<\) tells us that \(\frac{3}{4}\) is not part of the solution, meaning \(x\) must be less than \(\frac{3}{4}\).

By converting an inequality to interval notation and vice versa, you can better visualize and understand the set of values that the inequality describes.

Graphical Representation of Intervals

Graphical representation of intervals on a number line provides a visual means to understand and interpret the set of numbers within the interval. The number line uses points and lines to indicate which numbers are included in the interval and which are not.

In our exercise, to graphically represent the interval \([-\frac{1}{2}, \frac{3}{4})\), we:

• Draw a solid dot (or use a square bracket, depending on context) at \(-\frac{1}{2}\) to show inclusion of this endpoint.
• Draw an open circle (or use a round bracket) at \(\frac{3}{4}\) to indicate exclusion of this endpoint.
• Connect these two points with a solid line, representing the entire range of numbers from \(-\frac{1}{2}\) to just under \(\frac{3}{4}\).

This visual cue helps to easily interpret which parts of the number line are included in the interval. It's a practical way to verify or explore mathematical concepts, promoting better comprehension of numeric ranges and inequalities.