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Chapter 1: Problem 53

Graph the solution set and give the interval notation equivalent. \(x \leq 6\)

Short Answer

Expert verified
The solution is \((-\infty, 6]\).

Step by step solution

01

Identify the inequality

The given inequality is \( x \leq 6 \). This means that \( x \) takes on all values less than or equal to 6.
02

Determine the solution set

The solution set includes all real numbers \( x \) such that \( x \) is less than or equal to 6. This captures both negative numbers, zero, and all positive numbers up to and including 6.
03

Graph the inequality

On a number line, at the point \( x = 6 \), place a closed circle to denote that 6 is included in the solution set. Draw a line extending to the left from this point to indicate all numbers less than 6 are included.
04

Write the interval notation

The interval notation for the solution set of \( x \leq 6 \) is \((-\infty, 6]\). The parenthesis \((-\infty\) indicates \( -\infty \) is not a specific value we can reach, while the bracket \(6]\) shows 6 is included.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Solution Set
When dealing with inequalities, the solution set is the range of numbers that satisfy the inequality. In this case, the inequality \( x \leq 6 \) translates into a solution set that includes all real numbers less than or equal to 6. This means the solution contains lots of numbers: negative numbers, zero, and any positive number up to 6. The important part here is that 6 itself is included. Understanding that the solution set covers all these numbers helps in solving real-life problems where certain constraints are applied, such as ensuring that a maximum value is not exceeded.
Interval Notation Explained
The interval notation is a shorthand representation of the solution set in mathematical inequalities. It simplifies expressions of number ranges in a compact format. This is useful for equations like \( x \leq 6 \).
To express this in interval notation, we write \((-\infty, 6]\). Here's how it works:
  • The parenthesis \((-\infty\) means that the range extends indefinitely towards negative values, but not including any specific point like infinity since it's not a real number.
  • The bracket \(6]\) indicates that the number 6 is included in the solution set, aligning with the inequality's definition.
Understanding interval notation is crucial, as it provides a precise and concise way to describe the range of solutions in problems involving inequalities.
Graphing Inequalities on a Number Line
Graphing inequalities on a number line provides a visual representation of the solution set.
This makes it easy to see which numbers satisfy the inequality. Let's take our inequality \( x \leq 6 \) as an example:
  • Locate the point on the number line corresponding to 6. Place a closed circle on this number, which denotes that 6 is included in our set.
  • Draw a line extending indefinitely to the left from the circle. This line symbolizes all the numbers less than 6.
This simple graph clearly shows all permissible values for \( x \). Graphing inequalities helps to visually communicate solutions and makes it easier to understand constraints in mathematical or real-world problems.

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