Problem 45 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \([0, 0]\) on the number line, only including 0.

Step 1: Analyze the Inequality

The inequalities given are: \( x \leq 0 \) and \( x \geq 0 \). Both inequalities must be true at the same time.

Step 2: Find the Intersection

Since both conditions \( x \leq 0 \) and \( x \geq 0 \) must be satisfied at the same time, the only number that satisfies both is \( x = 0 \).

Step 3: Graph the Solution on a Number Line

On the number line, mark the single point \( x = 0 \). Use a solid dot to indicate that 0 is included in the solution.

Step 4: Write the Interval Notation

Since the only solution is the single number 0, the interval notation is \([0, 0]\). This represents the set containing only the number 0.

Key Concepts

Number Line

A number line is a simple visual representation of numbers along a straight line. It helps to give us a clear picture of inequalities and their solutions. By using a number line, we can visually interpret which numbers satisfy a given inequality.

When graphing solutions to inequalities, we use specific symbols:

• A solid dot indicates that a particular number is included in the solution.
• An open circle indicates that a number is not included.

For the inequality given in the exercise, where both conditions are \( x \leq 0 \) and \( x \geq 0 \), the only number satisfying both is 0. On the number line, this is depicted by a solid dot at 0, clearly showing 0 as our solution.

Intersection of Solutions

The intersection of solutions refers to the common solutions that satisfy all given conditions simultaneously. In this exercise, we have the inequalities \( x \leq 0 \) and \( x \geq 0 \). To find the intersection, we look for the numbers that make both inequalities true at the same time.

Let's break it down:

• For \( x \leq 0 \), any number less than or equal to 0 satisfies the inequality.
• For \( x \geq 0 \), any number greater than or equal to 0 satisfies the inequality.

The intersection, therefore, is the number that fits both criteria. In this case, the number 0 is the only solution, as it is the only number that is both equal to 0.

Interval Notation

Interval notation is a way of representing a set of numbers as an interval, using brackets to indicate whether endpoints are included. This notation is a concise and effective way to express solutions to inequalities.

For example:

• Brackets [ ] are used to show that the endpoints are included in the interval.
• Parentheses ( ) would be used to indicate that endpoints are not included.

In this exercise, where the solution is only the number 0, interval notation is written as \([0, 0]\). This indicates that the interval starts and ends at 0, containing only this single value. It's a neat way to show that the solution is precisely one number, without listing or trying to generalize. Thus, interval notation provides a compact way to express that the number 0 is both included and the only number in the set.