Problem 47 - Algebra and Trigonometry

Question

\(47-52\) : Express the inequality in interval notation, and then graph the corresponding interval. $$ x \leq 1 $$

Step-by-Step Solution

Answer

The inequality \(x \leq 1\) in interval notation is \((-\infty, 1]\) and it is graphed with a solid dot at 1 and shaded to the left.

Step 1: Identify the Inequality

The inequality given is \(x \leq 1\). This inequality describes the set of all numbers \(x\) that are less than or equal to 1.

Step 2: Express in Interval Notation

In interval notation, \(x \leq 1\) is expressed as \((-\infty, 1]\). This notation captures all numbers starting from negative infinity up to and including 1.

Step 3: Graph the Interval

To graph \((-\infty, 1]\), draw a number line. Place a solid dot at 1 to indicate that 1 is included, and shade the line to the left of 1, extending indefinitely to the left to represent \(-\infty\).

Key Concepts

Interval Notation

When working with inequalities, interval notation is a compact way to represent a range of numbers. It describes all possible solutions that satisfy the inequality. For the inequality \(x \leq 1\), interval notation is written as \((-\infty, 1]\).

• The "\((-\infty,1]\)" indicates that the solution set starts from negative infinity \(-\infty\) and goes up to 1, inclusive.
• Parentheses "()" are used for values that are not included in the set, while square brackets "[]" surround values that are included.
• Since infinity isn't a number we can reach, we always use an open parenthesis for it.
• The closed bracket "[1" shows that 1 is part of the solution.

This concise notation helps communicate the full range of solutions quickly and clearly.

Graphing Inequalities

To visualize inequalities, we graph them on a number line. This helps us see which numbers are included in the solution set. For the inequality \(x \leq 1\), follow these steps to create its graph:

• Draw a horizontal line to represent the number line.
• Locate the number 1 on this line and place a solid dot or circle at this point. This indicates that 1 is part of the solution.
• The inequality \(\leq\) means we include all numbers less than or equal to 1, so we shade the line to the left of 1.

Graphing inequalities always involves showing, at a glance, the span of numbers that satisfy the condition set by the inequality.

Number Line Representation

Number lines are graphical tools that help illustrate and solve inequalities. They offer a clear, visual understanding of where numbers fall in relation to each other.For the inequality \(x \leq 1\), the number line plays a crucial role in:

• Highlighting the Boundary: The number 1 acts as a boundary, marked clearly on the line, showing the transition in the solution set.
• Indicating Inclusion: A solid dot on the number 1 signifies it is included in the solution \(x \leq 1\), as opposed to an open dot for non-inclusive boundaries.
• Visualizing the Set: Shading to the left of the number 1, extending indefinitely, visually demonstrates the inclusion of all numbers less than or equal to 1, with the line continuing into infinity.

Using a number line makes understanding the scope of solutions simpler and more intuitive, emphasizing the concept of inclusion and exclusion in inequalities.