Question

Write an inequality that represents the statement and graph the inequality. \(x\) is less than 5 and greater than 2

Short answer

The inequality representing '\(x\) is less than 5 and greater than 2' is \(2 < x < 5\). A number line with open circles on 2 and 5 with a line drawn between them serves as the graphical representation.

Step-by-step solution

Write the Inequality

To represent \(x\) is less than 5 and greater than 2, an inequality can be written in this format: \(2 < x < 5\). This inequality indicates that \(x\) is exclusive of 2 and 5. If \(x\) is inclusive of 2 and 5, it would be written as \(2 \leq x \leq 5\). However, in this case, we take the former that excludes the boundary points.

Graph the Inequality

On the number line, points which represent 2 and 5 are marked. Since the values are not inclusive of 2 and 5, open circles are drawn on these points to indicate this. Lines are drawn between these points to represent the range of values \(x\) can take - all the values between 2 and 5.

Key concepts

Compound Inequalities

Compound inequalities involve two separate inequalities that are connected by the words 'and' or 'or'. For instance, when we say 'x is less than 5 and greater than 2', we are dealing with a compound inequality that combines two simple inequalities: 'x is less than 5' (\(x < 5\)) and 'x is greater than 2' (\(x > 2\)). The word 'and' indicates that both conditions must be true at the same time for a value to satisfy the compound inequality; in other words, only numbers between 2 and 5 (excluding 2 and 5 themselves) are solutions.

The compound inequality can be represented as a single statement, often in the format of 'a < b < c', where 'a' and 'c' are the lower and upper bounds, respectively. As such, the notation \(2 < x < 5\) effectively captures the range of numbers that x can be to satisfy both conditions of the inequality.

Number Line Representation

Graphing inequalities on a number line provides a visual representation of all possible solutions to the inequality. Each point on the number line corresponds to a real number, and the region represented by the inequality is shown using a line or ray.

When graphing compound inequalities like \(2 < x < 5\), we first mark the critical points - namely 2 and 5 in this case - on the number line. Open circles are used to denote that these endpoints are not included in the solution set (\textbf{exclusive} of 2 and 5), acknowledging that the inequality is strict (\textless and \textgreater). If the inequality included the endpoints, closed circles would be used instead to indicate inclusivity. For the solutions that fall between 2 and 5, a line is drawn connecting the open circles, visually representing all the numbers that x can possibly represent, which are the numbers between, but not including, 2 and 5.

Inequality Notation

Inequality notation is a system of symbols used to express the relationship between two quantities when they are not equal. The most common inequality symbols are \textless (less than), \textgreater (greater than), \textless= (less than or equal to), and \textgreater= (greater than or equal to). The symbol used indicates not only the relative size of quantities but also whether the boundary point (the specific value that makes the inequality an equation) is included in the set of solutions.

For instance, a strict inequality such as \(x < 5\) signifies that x can be any number less than 5, but cannot be 5 itself. Conversely, the notation \(x \textless= 5\) shows that x can be less than 5 and it can also be equal to 5. The importance of understanding these symbols cannot be overstated, as misinterpretation can lead to incorrect conclusions about the range of acceptable solutions. In the example given, \(2 < x < 5\), the 'less than' symbols convey a precise meaning: x can be any real number bigger than 2 and smaller than 5, with both 2 and 5 themselves excluded from the solution set.