Problem 9 - Algebra A Combined Function

Question

Graph the solution set of each inequality on a number line and then write it in interval notation. $$ \\{x \mid 5 \geq x>-1\\} $$

Step-by-Step Solution

Answer

The solution is graphed between -1 and 5, and the interval is \((-1, 5]\).

Step 1: Understand the Inequality

The inequality \( \{x \mid 5 \geq x > -1 \} \) defines a condition for the variable \( x \). Here, \( x \) is less than or equal to 5 and also greater than -1.

Step 2: Interpret the Inequality

Interpret the inequality to identify that \( x \) should satisfy two conditions: \( x \leq 5 \) and \( x > -1 \). These conditions define the interval of permissible values for \( x \) on the number line.

Step 3: Represent on a Number Line

Draw a number line, and mark the points for -1 and 5. For \( x > -1 \), use an open circle at -1 to indicate that -1 is not included. For \( x \leq 5 \), use a closed circle at 5 to show that 5 is included. Shade the region between these two points to represent all values \( x \) within the interval.

Step 4: Write in Interval Notation

Combine the conditions into interval notation. Since -1 is not included and 5 is included, represent this as \((-1, 5]\), where the parenthesis indicates that -1 is not included, and the bracket indicates that 5 is included.

Key Concepts

Number Line Representation

Visualizing an inequality on a number line can greatly enhance your understanding of the set of solutions. To graph the inequality \( \{x \mid 5 \geq x > -1 \} \), first draw a horizontal line. This line is your number line, a visual representation of all real numbers. Next, locate and mark the relevant points, which in this case are \(-1\) and \(5\). To indicate that \(x\) is greater than \(-1\), use an open circle around \(-1\). An open circle signifies that \(-1\) is not included in the solution set. Similarly, use a closed circle on \(5\). A closed circle indicates that \(5\) is indeed part of the solution set. Once your circles are in place, shade the area between them. This shaded segment on the number line shows all the numbers \(x\) that satisfy both conditions: greater than \(-1\) and less than or equal to \(5\). This graphical representation helps you see at a glance which values of \(x\) make the inequality true.

Interval Notation

Interval notation offers a concise way to express the set of solutions to an inequality. It's crucial to understand how open and closed intervals work. For the inequality \( \{x \mid 5 \geq x > -1 \} \), we use the interval notation \((-1, 5]\). Here's how to interpret it:

• Use a parenthesis \((\) for \(-1\) because the solution set does not include \(-1\). \((-1\) means \(x\) can be any number greater than \(-1\), but not \(-1\) itself.
• Use a bracket \([\) for \(5\) because the solution set contains \(5\). \(5]\) means \(x\) can be equal to \(5\).

So, \((-1, 5]\) captures all numbers between \(-1\) and \(5\), including \(5\) but excluding \(-1\). This notation is not only succinct but also universally understandable in mathematics.

Understanding Inequalities

Inequalities express a relation between two values when they are not equal but can be either greater than or less than one another. They describe a range rather than a fixed number. Let's deconstruct the inequality \( \{x \mid 5 \geq x > -1 \} \).

• First, this compound inequality combines two separate statements. It reads as "\(x\) is less than or equal to \(5\) and greater than \(-1\)."
• "\(x > -1\)" determines that all values of \(x\) are larger than \(-1\), but not \(-1\) itself.
• "\(x \leq 5\)" establishes that any value up to and including \(5\) is allowed.

Understanding inequalities involves recognizing these conditions and seeing how they define a continuous set of numbers that satisfy all parts of the inequality simultaneously. Inequalities are vital in describing feasible regions in many applications, from solving equations to optimizing real-world problems.