Question

Write an inequality that represents the statement and graph the inequality. \(x\) is less than 8 and greater than 3

Short answer

The inequality representing the statement is \(3 < x < 8\). The graph will have a line connecting open circles at 3 and 8, representing the range of solutions.

Step-by-step solution

Translate Statement into Inequality

The statement 'x is less than 8 and greater than 3' translates into the compound inequality \(3 < x < 8\). This means all the numbers that lie between 3 and 8 (not including 3 and 8 themselves) are solutions to this inequality.

Graph the Inequality

The graph is a number line with an open circle at 3 and 8 and a line connecting them. The open circles indicate that the numbers 3 and 8 are not included in the solution. The line between these two points represents all the possible solutions to the inequality, which are the numbers that lie between 3 and 8.

Key concepts

Compound Inequality

When we talk about a compound inequality, we're referring to a situation where there are two separate inequalities that we deal with at the same time. Think of it like a sandwich where two pieces of bread limit what can go in the middle—the values you can choose are 'sandwiched' between two restrictions.
The exercise presents a classic example of a compound inequality: you have two boundaries, 3 and 8, and the variable (in this case, \(x\)) can be any number that is more than 3 and less than 8. This type of inequality is often written with the smaller number on the left, the variable in the middle, and the larger number on the right, forming \(3 < x < 8\), as was revealed in the step-by-step solution. This notation is compact and shows the range of possible solutions for \(x\) in just one glance.
To make this concept a bit clearer, imagine a temperature gauge set to warn if the temperature is too hot or too cold. If the gauge turns on an alarm for temperatures below 3 degrees or above 8 degrees, then it is set to a compound inequality just like our exercise—alerts come for temperatures outside the 'safe zone' of 3 to 8 degrees.

Inequality Solution

An inequality solution is not just a single answer, as you would expect in a simple equation like \(x = 4\). Instead, it's often a whole range of possible answers, which can make it seem a bit trickier at the outset. In our exercise, the solution to the inequality \(3 < x < 8\) is any number greater than 3 but less than 8.
For example, \(4.5\), \(5\), and \(7.999\) are all solutions because they fall within the acceptable range. Hence, rather than pinpointing a single value, solving an inequality usually involves finding a set of values that make the inequality true. A tip to remember when working on inequalities is that they may have infinitely many solutions, as any number fitting the stated conditions should work. This is a mark of the flexible yet specific nature of inequalities—they define a continuum of values rather than a discrete point.

Number Line Graphing

The exercise demonstrates number line graphing, which is a visual way to represent the solution set of an inequality. A number line graph is particularly neat because it gives you a picture of all the numbers that satisfy the inequality.
When you're setting up a graph for a compound inequality like \(3 < x < 8\), you draw a number line and put markers at the key points—in this case, the numbers 3 and 8. Since the inequality does not include the endpoints, you use open circles to indicate that these numbers are not part of the solution set.
The line segment drawn between these open circles signifies all the numbers in between that are solutions. With number line graphing, you turn an abstract concept into something visually understandable.

Visualizing the Problem

If you imagine the numbers as points on a playground balance beam, you wouldn't be allowed to stand on the marks at 3 or 8 but anywhere in between.