Problem 80 - Algebra for College Students

Question

x<1

Step-by-Step Solution

Answer

The solution is all \( x < 1 \), written as \( (-\infty, 1) \) in interval notation.

Step 1: Understand the Inequality

The inequality we are working with is \( x < 1 \). This means that the value of \( x \) must be less than 1. Our goal is to find the values that satisfy this condition.

Step 2: Identify Possible Solutions

Considering the inequality \( x < 1 \), identify the range of values for \( x \). Since \( x \) must be less than 1, any real number that is smaller than 1 is a possible solution.

Step 3: Express the Solution on a Number Line

On a number line, indicate the point \( 1 \) with an open circle, indicating that \( x \) cannot be equal to 1. Shade the line to the left of \( 1 \), representing all numbers less than 1.

Step 4: Write the Solution in Interval Notation

In interval notation, the solution to the inequality \( x < 1 \) is expressed as \( (-\infty, 1) \). The parenthesis signifies that 1 is not included in the solution set.

Key Concepts

Number Line Representation

The number line is a graphical way to visualize inequalities like \( x < 1 \). It's a straight horizontal line with numbers placed at intervals. When we place an inequality on a number line, we show the range of values that satisfy the inequality. For \( x < 1 \), we use an open circle at point 1.

An open circle means that the point itself is not included in the solution. Since \( x \) must be less than 1, we shade the part of the line extending to the left of 1. This visually represents all numbers smaller than 1. A number line makes it easy to see which numbers fit the inequality and helps in understanding the solution better.

• Open circles denote that the endpoint is not included (\( < \) or \( > \)).
• Closed circles indicate the endpoint is included (\( \leq \) or \( \geq \)).

Interval Notation

Interval notation provides a concise way to express the range of numbers that satisfy an inequality. In interval notation for \( x < 1 \), we write the solution as \((-\infty, 1)\).

Here's how to understand it:

• The symbol \((-\infty,\) indicates there is no lower limit to the values \( x \) can take. They extend indefinitely leftward on the number line.
• The comma \(,\) separates the lower and upper boundaries of the interval.
• The parenthesis \(()\) next to 1 shows that 1 is not included in the set of solutions.

This notation is efficient and clear, making it simpler to communicate solutions, especially when complex inequalities are involved.

Inequality Solution Steps

Solving an inequality involves systematic steps to find the solution set. Consider the inequality \( x < 1 \).

Here are the steps, broken down:

• **Understand the Inequality**: Comprehend what the inequality is asking. \( x < 1 \) implies finding all numbers less than 1.
• **Identify Possible Solutions**: Think about numbers. By conceptualizing numbers, realize that any real number below 1 fits the criteria.
• **Graphical Representation**: Use a number line to clearly depict the solutions, using open circles and shading appropriately.
• **Express in Interval Notation**: For clarity and precision, articulate solution ranges such as \((-\infty, 1)\). This uniform method is widely recognized in mathematics.

By applying these steps, deriving solutions for inequalities becomes a manageable and clear process.