Problem 51 - Precalculus Mathematics for Calculus

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$x \leq 1$$

Step-by-Step Solution

Answer

The interval notation is \((-\infty, 1]\) and the graph is a ray with a filled circle at 1 extending leftwards.

Step 1: Understand the Inequality

The inequality given is \(x \leq 1\). This means that \(x\) can be any number less than or equal to 1. We need to express this range of values using interval notation.

Step 2: Convert to Interval Notation

In interval notation, inequalities are expressed as intervals. For \(x \leq 1\), \(x\) includes all numbers from negative infinity up to and including 1. The interval notation for this is \((-\infty, 1]\). We use a parenthesis for negative infinity because infinity is not a number that can be included, and a bracket for 1 to show that it is included in the interval.

Step 3: Graph the Interval

On a number line, the interval \((-\infty, 1]\) is represented by a line or ray starting from negative infinity and continuing through to 1. At 1, we use a filled circle to indicate that 1 is included in the interval.

Key Concepts

Inequalities

Inequalities are mathematical expressions used to compare two values or expressions. They tell us how one quantity relates to another. In general, inequalities can have symbols like:

• \(<\) which means 'less than'.
• \(>\) which means 'greater than'.
• \(\leq\) which stands for 'less than or equal to'.
• \(\geq\) which means 'greater than or equal to'.

For example, the inequality \(x \leq 1\) indicates that \(x\) can take any value that is less than or equal to 1. The inequality defines a range or set of possible values that \(x\) can take.

Understanding these relations helps us interpret the dataset or numbers involved. In real life, inequalities can signify limits or constraints, like temperatures not exceeding a certain level.

Graphing Inequalities

Graphing inequalities helps visualize the range of possible solutions on a number line or a coordinate system. This visual representation makes it easier to see the set of values that satisfy the given inequality. Here's how inequality \(x \leq 1\) could be graphed:

• Find the number 1 on your number line.
• Draw a point or circle at 1. As \(1\) is included in the solution set of \(x \leq 1\), you will fill in the circle.
• Shade all the numbers to the left of 1, extending towards negative infinity.

This shading indicates all possible solutions, as \(x\) could be any number less than or equal to 1. Graphing inequalities is a powerful tool that provides a visual understanding of mathematical expressions.

Number Line Representation

The number line is a simple method to represent the range of solutions for inequalities. Using a number line, we provide a visual cue that complements the abstract mathematical notations. To represent our example \(x \leq 1\):

• Locate the number 1 on the number line.
• Place a filled circle at 1, signifying that 1 is part of the solution set (because it's \(\leq\), 1 is included).
• Draw an arrow or a continuous line extending leftward from 1, indicating all the numbers less than 1, continuing indefinitely (towards negative infinity).

This visual aid helps students better understand which numbers satisfy the inequality. The simple act of shading or marking helps solidify the concept and bridges numerical understanding with visual interpretation.