Problem 20 - Introductory and Intermediate Algebra for College Students 4th

Question

Express the solution set of each inequality in interval notation and graph the interval. $$x<5$$

Step-by-Step Solution

Answer

The interval notation for the inequality \(x<5\) is \((-\infty, 5)\). The graph of the solution would feature an open circle at 5, with a line extending to the left from 5 towards negative infinity.

Step 1: Interpret the inequality

The inequality \(x<5\) means that we are looking for all values of \(x\) that are less than 5.

Step 2: Interval Notation

In interval notation, the solution for \(x<5\) is represented as \((-\infty, 5)\). Here, \(-\infty\) represents the lowest possible value for \(x\), which in this case is negatively infinite, while 5 is the upper limit, which isn't included in the solution set because of the '<' sign.

Step 3: Graphing

On a number line, begin by marking the number 5. Since 5 is not a part of the solution (as signified by the '<' sign, not '<='), we represent 5 with an open circle. Draw a line extending to the left of 5, towards negative infinity, showing all the numbers less than 5 are solutions. The open circle indicates that the point at 5 is not included in the solution.

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers along a number line. It’s particularly handy for showing the range of solutions to inequalities. For the inequality \(x<5\), you're interested in all numbers that are less than 5. To express this in interval notation, you write it as

• \((-\infty, 5)\)

The parenthesis "(" and ")" indicate that the endpoints are not included in the set. Thus, 5 is not part of the solution. The symbol \(-\infty\) is used to represent the idea that the solution extends indefinitely to the left on the number line.

Solution Set

The solution set of an inequality is the collection of all possible values that make the inequality true. For the inequality \(x<5\), the solution set includes every number that is less than 5.

• This can be anything like 4, 0, or even -100.
• However, it cannot include 5, since the inequality is strictly less than, not less than or equal to.

Using interval notation simplifies understanding and communication about these solutions, showing at a glance that every value below 5 is acceptable.

Graphing Inequalities

Graphing inequalities on a number line helps visualize which numbers are part of the solution set. For \(x<5\):

• Start by marking the number 5 with an open circle. The open circle indicates that 5 is not included in the solution.
• Draw a line to the left of 5, extending towards negative infinity, which shows all the numbers less than 5.

This graphical representation aids in understanding that every number to the left is part of the solution, providing a quick visual reference to accompany the interval notation.