Problem 44 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$\left[-\frac{3}{2}, \frac{1}{2}\right]$$

Step-by-Step Solution

Answer

The inequality notation is \(-\frac{3}{2} \leq x \leq \frac{1}{2}\); plot with closed circles at endpoints.

Step 1: Understand the Interval Notation

The interval \(\left[-\frac{3}{2}, \frac{1}{2}\right]\) is written in interval notation. Here, both endpoints, \(-\frac{3}{2}\) and \(\frac{1}{2}\), are included in the interval, meaning it is a closed interval.

Step 2: Convert Interval to Inequality Notation

To express the interval \(\left[-\frac{3}{2}, \frac{1}{2}\right]\) in inequality notation, we write it as \(-\frac{3}{2} \leq x \leq \frac{1}{2}\). This means that \(x\) can take any value starting from \(-\frac{3}{2}\) to \(\frac{1}{2}\) inclusive.

Step 3: Sketch the Interval on a Number Line

Draw a number line and mark the points \(-\frac{3}{2}\) and \(\frac{1}{2}\). Both points are closed circles, indicating they are included in the interval. Shade the region between these two points to show all the values \(x\) can take.

Key Concepts

Inequality Notation

In mathematics, inequality notation is used to express the range of values that a variable can take on. It often helps in representing intervals in a precise and compact form. When you see something like

• \(-\frac{3}{2} \leq x \leq \frac{1}{2}\)

this is an example of inequality notation. It signifies that \(x\), the variable, can be any number from \(-\dfrac{3}{2}\) up to \(\dfrac{1}{2}\), including both \(-\dfrac{3}{2}\) and \(\dfrac{1}{2}\).
Think of inequality notation as a way of writing down the boundary conditions for values of \(x\). This form is particularly useful when dealing with a range of values in functions or equations. It's a widely used notation because it specifies exactly what values are allowed without ambiguity.

Closed Interval

A closed interval is an interval that includes its endpoints. In mathematical terms, a closed interval from \(a\) to \(b\) is denoted as \([a, b]\). This means that all numbers between \(a\) and \(b\), including the numbers \(a\) and \(b\) themselves, are part of the interval.For the interval \([-\frac{3}{2}, \frac{1}{2}]\), both \(-\frac{3}{2}\) and \(\frac{1}{2}\) are included. The brackets \([ \ ])\) indicate that the interval is closed.
Closed intervals are particularly important when talking about continuous functions, ensuring no gaps on the endpoints. Mathematical problems often require factoring in these endpoints, and understanding whether they are included (closed interval) or not (open interval) is key to solving such problems.

Number Line Representation

Number lines are a straightforward way to visually show intervals and inequalities by placing them on a continuous line, usually with numbers spaced evenly. To represent an interval like \([-\frac{3}{2}, \frac{1}{2}]\) using a number line, you would do the following:

• Start by drawing a horizontal line.
• Mark the points \(-\frac{3}{2}\) and \(\frac{1}{2}\) on this line.

To indicate a closed interval, both endpoints \(-\frac{3}{2}\) and \(\frac{1}{2}\) are marked with solid, closed circles. This indicates these endpoints are included in the interval.
Next, shade the area between these two points. This shaded section reflects all the possible values that \(x\) could be within the interval. A number line helps in a quick understanding of ranges and is widely used in both arithmetic and algebra.