Question

Sketch the graph of the inequality. $$x<4$$

Short answer

The graph is a number line with an open circle at 4 and all points to the left of 4 shaded or highlighted.

Step-by-step solution

Draw a number line

First, you'll need to draw a horizontal line. This line represents all real numbers. The numbers increase as you move to the right on the line, and decrease as you move to the left.

Plot the value '4'

Indicate the point or value '4' on the number line. This is the number from your inequality.

Indicate the inequality

The inequality sign in your problem is '<', which means 'less than'. This suggests that the solution to the inequality includes all numbers less than 4. We should therefore draw an open circle at 4, indicating that 4 is not a part of the solution set, and then shade or highlight everything to the left of this point. This shaded region will represent all numbers less than 4 on the number line.

Key concepts

Number Line

A number line is a visual representation of numbers laid out in a straight horizonal path, with equally spaced intervals. It is an essential tool in mathematics for demonstrating various concepts, including inequalities. Each point on the line corresponds to a real number, and the position of numbers is based on their value; the further to the right, the greater the number, and vice versa. To graph an inequality on a number line, you first draw this line and then locate points that correspond to critical values from the inequality. Understanding the structure and purpose of the number line is fundamental to correctly portray mathematical relationships such as inequalities.

When working with a number line, remember to:

Inequality Notation

Inequalities are mathematical expressions showing the relative size or order of two values. They are represented by inequality signs such as '<' for 'less than', '>' for 'greater than', 'leq' for 'less than or equal to', and 'geq' for 'greater than or equal to'. It's important to choose the correct inequality symbol to accurately state the relationship between the numbers involved. For example, in our exercise, the notation '\(x<4\)' indicates that 'x' is any number less than '4', but does not include '4' itself.

Proper use of inequality notation is crucial for:

Open Circle on a Graph

In graphing inequalities on a number line, an open circle is used to represent a number that is not included in the solution set. This visual cue is crucial when differentiating between 'less than' and 'less than or equal to'. With the inequality ‘\(x<4\)’, we use an open circle at the value '4' to indicate that '4' is the boundary of our solution but is itself not a solution to the inequality. The open circle signals that we are considering all numbers up to, but not including, the number at which the circle is placed.

An open circle is a distinct symbol that means:

Shading the Number Line

Shading on a number line is a representation of all the values that satisfy an inequality. When the inequality is 'less than' or 'greater than', we shade all the numbers to the left (for less than) or right (for greater than) of the critical value up to the end of the number line in that direction. This creates a visual representation of the solution set and helps to quickly identify which numbers are included. In our exercise where \(x<4)\), we shade to the left of the open circle at '4' to represent all numbers less than '4'. It’s essential to shade accurately to convey the correct set of solutions. Shading is important for: