Question

In Problems 31-34, graph the numbers \(x\) on the real number line. \(x \leq 7\)

Short answer

Graph includes a closed circle at 7 with shading extending to the left.

Step-by-step solution

- Understand the inequality

The inequality given is \( x \leq 7 \). It means we are looking for all real numbers \( x \) that are less than or equal to 7.

- Draw the number line

Draw a horizontal line and mark several points on it to represent different values. Ensure that the point 7 is clearly marked since it is crucial to this inequality.

- Identify the region

Since \( x \leq 7 \), we include all numbers to the left of 7 as well as 7 itself. To show this, we use a closed circle (or filled-in dot) at 7 and shade the region to the left of the point, extending indefinitely.

- Graph the inequality

Place a closed circle at 7 on the number line to indicate that 7 is included. Shade the entire region to the left of the number 7 to show all numbers that are less than or equal to 7.

Key concepts

Real Number Line

When graphing inequalities, we use the real number line to visually represent the range of values that satisfy the inequality. A real number line is simply a horizontal line with numbers placed at equal intervals. This line extends infinitely in both directions, capturing all possible real numbers from negative infinity to positive infinity.

To make it easier to understand inequalities, it's useful to mark key points on the number line. In this exercise, the point 7 is a critical value, so it should be clearly labeled.

Inequality Graphing

Inequality graphing helps us visualize which parts of the real number line satisfy a given inequality. For instance, the inequality \( x \leq 7 \) tells us that the solution includes all numbers less than or equal to 7.

When graphing such inequalities, you must first understand the inequality symbol:

• '≤' means 'less than or equal to'.
• '>' means 'greater than'.

For \( x \leq 7 \), we need to include the number 7 and all values less than 7. This step is crucial as it ensures we accurately represent the inequality on the real number line.

Number Line Shading

Shading on a number line visually represents all the numbers that satisfy an inequality. For inequalities like \( x \leq 7 \), shading will be crucial to clearly indicate the solution region.

In this example:

• Begin by drawing a closed (filled-in) circle at 7. This shows that the point 7 is included in the solution.
• Next, shade the line to the left of 7. This shading must continue indefinitely to the left to represent all numbers less than 7.

This approach makes it easy to understand which values are part of the solution and which are not.

Included Values

Whenever you are working with inequalities and number lines, it's essential to understand which values are included in the solution set. In this problem, the key point is that 7 itself is part of the solution.

To indicate such included values:

• Use a closed circle (filled-in dot) on the number line at the point of inclusion.
• If the point isn't included in the solution, you would use an open circle instead.

For \( x \leq 7 \), we include the value 7, so we use a closed circle at 7 and shade everything to the left. This practice ensures a clear and accurate representation of the inequality.