Problem 41 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$(2,5)$$

Step-by-Step Solution

Answer

The interval \((2, 5)\) as an inequality is \(2 < x < 5\), represented with open circles at 2 and 5 on a number line.

Step 1: Understanding the Interval

The given interval is \((2, 5)\), which is expressed using parentheses. Parentheses indicate an open interval, meaning that the endpoints 2 and 5 are not included in the interval.

Step 2: Expressing the Interval as an Inequality

To express the interval \((2, 5)\) as an inequality, we consider all the values \(x\) such that 2 is less than \(x\), and \(x\) is less than 5. This can be written as the inequality: \(2 < x < 5\).

Step 3: Graph the Interval on a Number Line

To represent the interval \((2, 5)\) on a number line, draw a line and plot points at 2 and 5. Since the interval is open, use open circles at 2 and 5 to indicate these points are not included. Shade the line between them to show all numbers greater than 2 and less than 5 are included.

Key Concepts

Open Intervals

Intervals help us understand a range of numbers between two endpoints. An open interval means these endpoints are not part of the interval. So when you see something like \((2, 5)\), it tells us that the numbers in the range begin just after 2 and end just before 5.

These are denoted using parentheses, like \((...)\), to clearly show that the endpoint numbers are not included. This is different from closed intervals, which use square brackets \([...]\) to indicate that the endpoints are included.

Open intervals are especially useful in scenarios where the exact boundary values are not relevant but all numbers in between are considered. They are common in areas like calculus or when studying continuous functions.

Inequality Expressions

Inequality expressions help us describe intervals mathematically. To turn an open interval into an inequality, we replace the endpoints with less than signs. For instance, the open interval \((2, 5)\) is expressed as the inequality \(2 < x < 5\).

This notation means any number \(x\) that is greater than 2 and less than 5 satisfies the inequality. Essentially, inequalities help us zero in on precise conditions that a number must meet to be part of the described set.

Inequalities can seem tricky at first, but they’re just a way of expressing these conditions clearly and mathematically. Make sure to pay attention to whether the inequality involves less than (\(<\)) or less than or equal to (\(\leq\)). This slight difference reflects whether the boundaries are open or closed.

Graphing on a Number Line

A number line represents numbers on a straight line, which helps visualize intervals. To graph an open interval like \((2, 5)\), start by identifying the endpoints 2 and 5 on the line.

Since these endpoints are not included in the interval, mark them with open circles. An open circle shows that the exact number at that point is not part of the interval.

Next, shade the line between these open circles. This shaded area represents all numbers that are part of the interval. It’s like showing that any number you pick from this shaded region belongs to the interval \(2 < x < 5\).

Graphing on a number line allows you to visually communicate the range of values you are dealing with, making it easier to understand and interpret different mathematical concepts.