Problem 23 - Intermediate Algebra

Question

Graph each inequality. $$ x<4 $$

Step-by-Step Solution

Answer

Graph a line with an open circle at 4 and shade left of 4.

Step 1: Understand the Inequality

The inequality is given as \( x < 4 \). This means that we need to consider all values of \( x \) that are less than 4.

Step 2: Draw the Number Line

Draw a horizontal line and mark relevant numbers, including the number involved in the inequality (4 in this case). Include numbers less than 4, such as 3, 2, and 1, on the left, making sure there is enough space to represent these values.

Step 3: Indicate the Boundary

Locate the number 4 on the number line. Since the inequality is strict (\( < \), not \( \leq \)), you must represent the boundary at 4 with an open circle. This shows that 4 itself is not included in the solution set.

Step 4: Shade the Solution Set

Shade the part of the number line to the left of 4, extending the shading through all values that are less than 4, indicating that these are the solutions to the inequality \( x < 4 \). Ensure the shading does not include the open circle at 4.

Key Concepts

Understanding the Number Line

A number line is a simple tool used to visually represent numbers. It consists of a straight horizontal line where each point corresponds to a number. To construct a number line:

• Choose a start point, often zero, and mark it clearly.
• Decide the scale and spacing between the numbers, ensuring equal intervals.
• Mark significant numbers involved in your problem, like key points to show boundaries and solutions.

For example, in the inequality problem with \( x < 4 \), we focus on the number line covering numbers less than 4. Placing 4 and other numbers such as 3, 2, and 1 helps to see the range of solutions. The number line is like the backbone for visualizing our solution in cases of inequality problems.

Solving and Graphing the Inequality Solution

Graphing an inequality is about finding all the possible solutions that satisfy the inequality condition. Here, we're interested in \( x < 4 \). To solve it and represent it graphically:

• Identify that numbers less than 4 are part of the solution. This includes numbers like 3, 2, 1, and so on, even those approaching up to 4 but not including it.
• On the number line, this is shown by shading the section to the left of 4.
• The shading extends infinitely towards the left, as smaller and smaller values meet the inequality condition of being below 4.

Remember, graphically representing the inequality helps us quickly see all possible numbers (solutions) that keep \( x < 4 \) true.

Representing Boundaries on a Number Line

When graphing inequalities, how we indicate the boundary is crucial. The boundary tells us where the solutions begin or end. In \( x < 4 \), the number 4 is the boundary, showing that no solution is larger or equal to 4:

• Since the inequality uses \( < \) (less than), 4 itself is not part of the solution set. We represent this using an open circle on the number line.
• An open circle at 4 visually communicates that 4 is the cutoff point, not included among the solutions, and the shading starts immediately beside it but does not cover it.
• Open and closed circles are standard ways to mark boundaries in graphing inequalities, providing clear interpretation visible through graphical representation.

This clear boundary indication is essential for accurately reading and understanding the graph of an inequality.