Problem 32 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x<12 \text { or } x>-6 $$

Step-by-Step Solution

Answer

The solution in interval notation is (-6, 12).

Step 1: Analyze the Inequalities Separately

There are two separate inequalities to consider: 1. \( x < 12 \) 2. \( x > -6 \) These inequalities describe two sets of solutions on the number line.

Step 2: Graph Each Inequality

For \( x < 12 \), draw an open circle at 12 and shade the number line to the left to indicate all numbers less than 12. For \( x > -6 \), draw an open circle at -6 and shade the number line to the right to indicate all numbers greater than -6.

Step 3: Combine the Solutions

Since the solutions form an 'or' inequality, any number that satisfies at least one of the inequalities is part of the solution set. Thus, draw a number line with combined shading from both previous steps: - From the open circle at -6, shade all numbers to the right, combining with the shading for numbers less than 12.

Step 4: Write the Solution in Interval Notation

In interval notation, the solution is written by combining both intervals. The numbers range from all numbers greater than -6 to numbers less than 12. In interval notation, this is written as: \((-6, 12)\).

Key Concepts

Number Line

A number line is a visual tool that helps us understand inequalities and their solutions. It is a straight line that displays numbers in increasing order from left to right. When graphing inequalities on a number line, we represent solutions with shaded regions and specific symbols.

For inequalities like \(x < 12\) or \(x > -6\), you will:

• Draw the line with an appropriate scale.
• Plot points for the boundary values. Here, you place points at 12 and -6.
• Use arrows to indicate extending solutions beyond visible values.

These steps help visualize which sections of the number line represent the solutions to the inequalities.

Interval Notation

Interval notation is a concise way to describe a range of numbers, often used to express the solutions of inequalities. This method uses brackets to show whether endpoints are included or not.

For the inequality \(x < 12\) or \(x > -6\), the solution is the range of numbers greater than -6 and less than 12. We write this as \((-6, 12)\) in interval notation.

• Parentheses \(()\) mean the endpoint is not included.
• Brackets \([]\) mean the endpoint is included.

In our case, because neither -6 nor 12 is included (marked by open circles), we use parentheses.

Open Circle

The open circle is an important symbol used on number lines to denote that a point is not included in the solution set of an inequality.

When you see a statement like \(x < 12\) or \(x > -6\), they indicate that 12 and -6 themselves are not part of the solution.

To use open circles:

• For \(x < 12\), place an open circle at 12. This shows values approaching 12, not 12 itself.
• For \(x > -6\), place an open circle at -6, indicating values greater than -6 but not -6 itself.

This concept ensures clarity in which numbers are possible solutions, avoiding any confusion about boundary inclusion.