Problem 44 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 5\) and in interval notation: \([5, 5]\).

Step 1: Understand the Inequality

The exercise involves two inequalities: \(x \leq 5\) and \(x \geq 5\). Both of these inequalities require \(x\) to be 5, since it cannot be greater or lesser.

Step 2: Finding the Common Solution

Since the inequalities \(x \leq 5\) and \(x \geq 5\) intersect at the single point where \(x\) equals 5, the only solution is \(x = 5\).

Step 3: Represent on a Number Line

To graph the solution on a number line, plot a solid dot at the point where \(x = 5\). This indicates the specific value included in the solution.

Step 4: Interval Notation

In interval notation, the solution \(x = 5\) is represented by the single point \([5, 5]\), denoting that the value is only 5.

Key Concepts

Number Line Graph

A number line graph is a visual representation of numbers along a straight line. In the context of inequalities, it helps us see which values satisfy a given condition.
When graphing inequalities, you place dots or circles on the number line to show the values that are part of the solution set.

• Solid Dot: Use a solid dot for values that are included in the solution, such as in "less than or equal to" (\(\leq\)) or "greater than or equal to" (\(\geq\)) inequalities.
• Open Circle: For values that are not included, like those only "less than" (<) or "greater than" (>), use an open circle.

In this exercise, because of the equality present in both conditions \(x \leq 5\) and \(x \geq 5\), the point \(x = 5\) on the number line is represented with a solid dot. This dot at 5 clearly shows that 5 is the only number that satisfies the given inequalities. The number line aids in easily identifying solution sets, especially when dealing with more complex inequalities.

Solution Set

The term solution set refers to the collection of all possible solutions that satisfy an equation or inequality. In simple terms, it answers the question: What values of the variable make the inequality true?
In our example, the inequalities \(x \leq 5\) and \(x \geq 5\) simplify to \(x = 5\). Therefore, the solution set here is just the single number 5.

• Simplifying helps in breaking down complex inequalities to easily digestible parts.
• Identifying the solution set allows you to clearly state which values meet the inequality's conditions.

When working with inequalities, always look for overlaps or intersections of multiple inequalities to find the common solution set. The solution set concept is vital as it tells you exactly which values of the variable are viable, guiding you through solving related mathematical problems.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is especially useful for describing solution sets of inequalities and represents them concisely.
The notation uses brackets "[ ]" and parentheses "( )" to show which numbers are included or excluded:

• [a, b]: Both a and b are included.
• (a, b): Both a and b are excluded.
• [a, b): a is included, but b is not.
• (a, b]: a is not included, but b is.

In this exercise, the inequality \(x = 5\) is depicted as \([5, 5]\), meaning only the number 5 is part of the solution set. No interval exists beyond this point, as the solution can't extend beyond the single included value. Interval notation is straightforward, yet crucial for conveying the range of solutions efficiently in a streamlined format.

Elementary Algebra

Elementary algebra forms the foundation for more advanced math. It typically involves understanding and solving equations and inequalities, like those presented in this exercise.
Key concepts include:

• Variables: Symbols, like x, that stand for unknown values.
• Equations and Inequalities: Mathematical statements that show the relationship between expressions.
• Simplification: Reducing expressions to simpler forms for easier analysis.

Algebra teaches you how to manipulate expressions to find solutions, as seen with the simple combination of \(x \leq 5\) and \(x \geq 5\) into \(x = 5\). This exercise demonstrates how elementary algebra can take statements about numbers and turn them into clear, logical conclusions. Understanding this foundation is critical for tackling more complicated problems in mathematics.