Problem 5 - Intermediate Algebra

Question

Express the given inequality in interval notation and sketch a graph of the interval. \(x<-2\)

Step-by-Step Solution

Answer

The interval is \((-\infty, -2)\), shown with an open circle at \(-2\) and shaded to the left.

Step 1: Understand the Inequality

The given inequality is \(x < -2\). This means that \(x\) includes all numbers that are less than \(-2\). The inequality is a strict inequality, indicating that \(-2\) itself is not included.

Step 2: Convert to Interval Notation

Since \(x\) includes all numbers less than \(-2\) but not \(-2\) itself, we express this using interval notation as \((-\infty, -2)\). Intervals with infinity are always open intervals and the interval is opened on negative 2 as well, since negative 2 is not included.

Step 3: Sketch the Graph

To graph the interval \((-\infty, -2)\), draw a number line. Place an open circle at \(-2\) to indicate that \(-2\) is not included in the interval. Then shade the line to the left of \(-2\) to represent all numbers less than \(-2\). The open circle signals the end point is not part of the interval.

Key Concepts

Inequality Graphing

Graphing an inequality like \(x < -2\) helps visualize the solution set on a number line. To represent this inequality correctly:

• First, identify the critical value where the change occurs, which is \(-2\) in this case.
• Since it’s a strict inequality \((<)\), \(-2\) is not part of the solution, so we use an open circle to clearly highlight this.
• Next, shade the line extending to the left of \(-2\) to indicate all values less than \(-2\) satisfy the inequality.

Graphing inequalities aids in understanding which numbers satisfy a condition, providing a visual tool alongside the mathematical representation.

Number Line

A number line is a visual representation aiding in the clear understanding of numbers and inequalities. When sketching a number line for inequalities:

• Draw a horizontal line with evenly spaced markings representing different numbers.
• Identify the crucial points relevant to your inequality - here it's \(-2\).
• An open circle is placed on the number line at critical points not included in the solution, such as at \(-2\) for \(x < -2\).
• Shade the area on the line that represents the solution. For this < inequality, shade to the left of the open circle.

Number lines provide a straightforward method to see solutions of inequalities and ensure you correctly interpret the math.

Strict Inequalities

Strict inequalities, such as \(x < -2\), are a type of inequality where the number on the boundary is not part of the solution set. Unlike non-strict or inclusive inequalities (\(\leq\) or \(\geq\)), strict inequalities:

• Do not include the boundary number, meaning for \(x < -2\), \(-2\) isn’t included.
• Represented graphically, the non-inclusive nature is shown with an open circle.
• Interval notation for strict inequalities always uses parentheses, not brackets, as is the case with < \((-< \infty, -2)\).

Understanding strict inequalities is crucial for correctly interpreting and communicating mathematical concepts in both written and graphical forms.