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Chapter 4: Problem 55

Graph each inequality. Then describe the graph using interval notation. $$ x<4 $$

Short Answer

Expert verified
The graph is an open interval \((-\infty, 4)\).

Step by step solution

01

Understanding the Inequality

The inequality given is \( x < 4 \). This means we are looking for all values of \( x \) that are less than 4. The inequality doesn't include the number 4 itself.
02

Sketching the Graph

On a number line, draw an open circle at \( x = 4 \) to show that 4 is not included. Shade all the values to the left of 4, indicating all numbers less than 4.
03

Describing the Graph in Interval Notation

In interval notation, an inequality of \( x < 4 \) is represented as \((-\infty, 4)\). The parentheses indicate that 4 is not included in the solution set, and infinity always uses a parenthesis.

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Key Concepts

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

Number Line
A number line is a visual representation of numbers placed at equal intervals along a straight line. It helps in plotting and understanding the range of solutions to inequalities.To graph the inequality \(x < 4\) on a number line, begin by drawing a horizontal line with evenly spaced numbers. Include a point for the number 4 as it's a critical value in this context. However, since the inequality is less than, 4 itself is not included in the solutions.To indicate this on the number line:
  • Draw an open circle at the position where 4 is located, emphasizing that 4 is not a solution.
  • Shade or extend the line to the left of 4, capturing all the numbers that are actual solutions (i.e., numbers less than 4).
This visual approach clarifies which numbers satisfy the inequality and help you understand the solution set intuitively.
Interval Notation
Interval notation provides a concise way to represent a set of numbers, particularly when dealing with continuous ranges like those found in inequalities.For the inequality \(x < 4\):
  • The range of solutions includes all numbers less than 4, but not 4 itself.
  • In interval notation, this is written as \(( -\infty, 4 )\).
Here's what each part of this notation means:
  • The parenthesis \((\) means that the number 4 is not included. We use a round bracket for numbers not included in the set.
  • The "minus infinity" symbol \(-\infty\) signifies that the range extends indefinitely to the negative side of the number line.
  • It uses parentheses because infinity is not a reachable or defined number that can be included.
Understanding interval notation helps you compress information about ranges into a compact and tidy mathematical expression without losing clarity.
Inequality Representation
Inequalities express a relationship between two values where one is not necessarily equal to the other. Instead, it shows how one value relates by being greater or lesser than the other.For the inequality \(x < 4\):
  • The symbol "<" denotes "less than," indicating that all numbers fulfilling this inequality are smaller than 4.
  • Unlike equations, inequalities offer a range of solutions rather than just one specific value.
When graphing an inequality:
  • Use an open circle to highlight numbers not included (as in \(x < 4\), the number 4 itself is excluded).
  • The graph on the number line extends in a direction (left for less than, right for greater than) to illustrate all potential solutions.
Inequalities are less rigid compared to equations鈥攖hey allow for flexibility in maintaining relationships, making them a valuable tool in various mathematical contexts.

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