Problem 40 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x \geq 0 \text { and } x<5 $$

Step-by-Step Solution

Answer

The interval is \([0, 5)\).

Step 1: Analyze the Inequalities

The problem provides two inequalities: \( x \geq 0 \) and \( x < 5 \). This tells us that we are looking for numbers \( x \) that are greater than or equal to 0 and also less than 5.

Step 2: Combine the Inequalities

When combining the inequalities \( x \geq 0 \) and \( x < 5 \), we identify the range where both conditions are satisfied. Therefore, \( 0 \leq x < 5 \) is the solution set for the range that meets both criteria.

Step 3: Graph the Solution on a Number Line

Draw a number line. Place a closed circle at \( x = 0 \) to indicate that 0 is included in the solution set. Place an open circle at \( x = 5 \) to indicate that 5 is not included in the solution set. Shade the region between \( 0 \) and \( 5 \) to show all numbers in this range.

Step 4: Write the Interval Notation

The solution \( 0 \leq x < 5 \) in interval notation is \([0, 5)\). The square bracket indicates that 0 is included, while the parenthesis indicates that 5 is not included.

Key Concepts

Number Line

A number line is a simple way to visually represent numbers. It serves as a graphical representation of numerical data, helping you to understand inequalities better. To represent a solution set on a number line, you start by drawing a horizontal line.
Mark points on the line that correspond to the important boundary numbers of the inequality, in this case, 0 and 5.

• Use a closed circle at a number if that number is included in the solution set, like at 0 for \(x \geq 0\).
• Use an open circle if the number is not included, such as at 5 for \(x < 5\).

Once you have marked these points, you continue by shading the region between these points to indicate it contains all the solutions to the inequality. This shading shows the solutions entirely, making it clear and complete.

Interval Notation

Interval notation is a concise way of expressing sets of numbers, particularly useful for representing ranges of solutions in inequalities. In our problem, the interval notation for the solution set \(0 \leq x < 5\) is written as \([0, 5)\).

• The square bracket \([\) at 0 signifies that 0 is part of the solution set, meaning "inclusive." This comes from the "\( x \geq 0 \)" condition.
• The parenthesis \()\) at 5 denotes that 5 is not included, referred to as "exclusive," aligning with the "\( x < 5 \)" condition.

This notation is very efficient for mathematicians and students as it's both compact and precise, offering a quick snapshot of which numbers belong to the solution set.

Solution Set

The solution set encompasses all the values of \(x\) that satisfy both inequalities in the exercise: \( x \geq 0 \) and \( x < 5 \). When a problem involves multiple inequalities, you must find a range where all individual conditions overlap.
In this case, both inequalities combine to form "\( 0 \leq x < 5 \)." This means any number starting from 0 up to but not including 5 will be part of our solution set.

• This is shown on the number line as a portion highlighted between 0 and 5.
• In interval notation, this is recorded as \([0, 5)\).

Understanding the solution set is crucial for solving inequality problems effectively as it returns the applicable range of solutions.