Question

In Problems 31-34, graph the numbers \(x\) on the real number line. \(x<4\)

Short answer

Shade the number line to the left of an open circle at 4.

Step-by-step solution

- Understand the inequality

The inequality is given as \( x < 4 \). This means we are looking for all real numbers that are less than 4.

- Determine the critical point

Identify the critical point from the inequality, which is the number 4. This is the point where the inequality changes from true to false.

- Draw the number line

Draw a horizontal line and mark points on it to represent the number line, including the critical point 4.

- Represent the critical point

Since the inequality is strictly less than (\( < \)) and does not include 4, use an open circle to represent 4 on the number line.

- Shade the appropriate region

Shade the region to the left of the open circle at 4, extending infinitely to the left. This shaded region represents all real numbers less than 4.

Key concepts

Real Number Line

The real number line is a visual tool used in mathematics to represent all real numbers. It's essentially a horizontal line that extends infinitely in both directions. Points on this line correspond to real numbers, increasing toward the right and decreasing toward the left.
When graphing inequalities, the real number line helps to easily visualize the set of solutions. To start, mark significant points; in this case, we are particularly interested in 4.

Inequality Representation

An inequality shows the relationship between two expressions. Here, we have the inequality ewline \( x < 4 \). This means that the value of \( x \) is always less than 4. It tells us that \( x \) can take any value that is not 4 but is smaller than 4.

When graphing this inequality on the real number line, it's important to understand what the symbols represent. The \( < \) sign indicates that the value is strictly less than 4 and does not include 4 itself. This helps us know where to make our markings and what kind of circles (open or closed) to use.

Critical Points

Critical points are key values that separate regions of the number line where an inequality holds true or false. For \( x < 4 \), the critical point is 4. This is where the inequality condition changes.

When we plot the critical point on the real number line, we need to mark it carefully. Since our inequality is strictly less than 4, we use an open circle at this point. An open circle indicates that 4 itself is not included in the solution set.

Shading Regions

The final step in graphing an inequality involves shading the regions of the number line that satisfy the inequality. For \( x < 4 \), we need to shade the entire region to the left of the open circle at 4.

This shaded area shows all the real numbers that are less than 4, extending infinitely to the left. Anytime you see a shaded region on a number line, it's indicating the range of values that satisfy the inequality.

By correctly shading, marking critical points, and understanding the inequality symbols, you can effectively visualize and solve inequalities on the real number line.