Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

Express the solution set of each inequality in interval notation and graph the interval. $$x<4$$

Step-by-Step Solution

Answer

The solution set of the inequality \(x < 4\) in interval notation is \((-\infty, 4)\). This is represented on a number line by an open circle at 4 with everything to its left shaded.

Step 1: Understanding the inequality

The inequality given here, \(x < 4\), represents a relationship where \(x\) is any real number that is less than 4.

Step 2: Interval notation

We use interval notation to represent a range of numbers. For the given inequality \(x < 4\), the interval notation will be \((-\infty, 4)\). This means it includes all real numbers less than 4, but does not include 4 itself.

Step 3: Graphing the result

On a number line, we can represent the solution set by drawing an open circle at 4 (because 4 is not included in the solution set) and shading everything to the left. This shows that all numbers less than r are part of the solution set.

Key Concepts

Interval Notation

Interval notation is a concise way to represent sets of numbers, particularly intervals on the real number line. It indicates the start and end of an interval, specifying whether any boundary numbers are included.- Parentheses \(( )\) are used to denote endpoints that are not included in the interval, known as open intervals.- Square brackets \([ ]\) indicate endpoints that are included, known as closed intervals.For example, the inequality \(x < 4\) is expressed in interval notation as \(( -\infty, 4 )\). Here, infinity \(( \infty )\) signifies that there is no boundary in one direction, since we can't identify a largest or smallest number in the real number scale.Remember:

• Use \(( \infty )\) or \(( -\infty )\) with parentheses only, as infinity is a concept rather than a number and cannot be included.
• Check endpoints carefully to decide whether to use parentheses or brackets.

These signs provide clear information about the values contained in the interval and are essential when communicating mathematical ideas.

Graphing Inequalities

Graphing inequalities on a number line is a visual way to represent all the solutions of an inequality. When graphing, there are key steps to consider to make your graph accurate and easy to understand.To graph the inequality \(x < 4\) on a number line:

• Locate the number 4 on the number line. Since \(x\) is less than 4, shade everything to the left of this point to indicate all possible values for \(x\).
• Place an open circle (or dot) at 4, because 4 is not included in the solution set. This circle visually indicates that the boundary point is not part of the solutions.

Reading and drawing these graphs:
- Open circles remind us that the point itself is excluded.
- Shading in one direction shows all the numbers included in the inequality solution.Graphing inequalities helps in visualizing solutions and makes it simpler to understand the relationship between numbers.

Real Numbers

Real numbers are a broad category that includes almost all number types we use in everyday mathematics. This continuous number set includes integers, fractions, and irrational numbers.Real numbers can be:

• Positive numbers like \(1, 2.5, \text{or } \pi\).
• Negative numbers such as \(-3, -rac{7}{4}, \text{or } -\sqrt{2}\).
• Whole numbers and fractions \((\text{such as } \frac{1}{2} \text{ or } -\frac{11}{3})\).
• Zero \((0)\) is also a real number.

In the context of inequalities, when we say \(x < 4\), \(x\) represents any real number less than 4. It means all numbers below 4 in the real number system are included. The flexibility and infinite nature of real numbers make them essential in expressing and solving inequalities. All these qualities enable mathematical expression and problem-solving across numerous applications and contexts.