Question

Which of the following graphs shows all the possible numbers represented by the inequality \(30<15 x \leq 90\).

Short answer

To solve the inequality \(30 < 15x \leq 90\), first divide all parts by 15, resulting in the interval \(2 < x \leq 6\). The graph representing this interval has an open circle at x = 2 and a closed circle at x = 6, with shading between them. Find the graph that matches this description.

Step-by-step solution

Solve the inequality for x

To find the interval for x, we need to solve the inequality by dividing both sides of the inequality by 15. \[ \frac{30}{15} < \frac{15x}{15} \leq \frac{90}{15} \] This simplifies to: \[ 2 < x \leq 6 \]

Describe the interval

Now, we know that the possible values for x are in the interval (2, 6] which means x is strictly greater than 2 but less than or equal to 6.

Determine the graph

The graph that represents the interval (2, 6] will be a number line with an open circle at x=2 (indicating that x is strictly greater than 2, so not equal to 2) and a closed circle at x=6 (indicating that x can also be equal to 6). In between the open and closed circles, the graph should be shaded to show that all the numbers within the interval are included. Based on this description, identify the graph that corresponds to the solution of the inequality \(30 < 15x \leq 90\).

Key concepts

Inequality Number Line

Understanding how to represent inequalities on a number line is crucial for visualizing the set of solutions to an algebraic inequality. In the example given, the inequality to solve is \(30 < 15x \leq 90\). After solving, we find that the solution is \(2 < x \leq 6\). To represent this solution on a number line, we need to graph this interval.

The process begins with a number line which is a visual representation of numbers laid out in order on a straight line. To indicate that \(x\) is strictly greater than 2 but less than or equal to 6, we draw an open circle at 2 and a closed circle at 6. The open circle signifies that the number 2 is not included in the solution set, while the closed circle indicates that 6 is included. We shade or color in the line between the open circle at 2 and the closed circle at 6, which communicates that all the numbers in this range are part of the solution set.

An inequality number line makes it easy to see which numbers satisfy the inequality and helps establish a better understanding of solution sets in inequality problems.

Algebraic Inequalities

Algebraic inequalities play a significant role in mathematical concepts and real-world applications. They are expressions that show a relationship of inequality—less than, greater than, less than or equal to, or greater than or equal to—between two algebraic expressions. In our exercise, the algebraic inequality is \(30 < 15x \leq 90\).

To solve it, the inequality is manipulated much like an equation, with careful attention to the rules concerning the multiplication or division of negative numbers, which can reverse the inequality sign. Solving \(30 < 15x \leq 90\) requires dividing each part of the inequality by 15, which maintains the inequality and isolates \(x\). The result is \(2 < x \leq 6\), expressing that \(x\) takes on values that are larger than 2 and up to and including 6.

• When \(x\) is less than or greater than a certain value, use '<' or '>' signs respectively.
• When \(x\) can also be equal to the boundary values, use '\leq' or '\geq' signs accordingly.

Students are often required to solve and graph these inequalities to find the range of possible solutions for \(x\).

Interval Notation

Interval notation is a shorthand way of writing sets of numbers, often used to represent the solution sets for inequalities. In our example, after solving the inequality, we write the solution as \(2 < x \leq 6\), which corresponds to the interval notation (2, 6]. Interval notation includes a set of numbers between two endpoints, where the use of parentheses '()' indicates that the endpoint is not included, and the use of brackets ']' or '[' signifies that the endpoint is included.

In interval notation,

• Parentheses (2, ... ) indicate that 2 is not in the set;
• Brackets [... , 6] show that 6 is included;

This notation is concise and clear, showing the boundary numbers and whether they are part of the solution set. It is essential in communicating the precise range of solutions to algebraic inequalities, especially when they are represented graphically on a number line.

Understanding interval notation is a fundamental step in mastering algebra and calculus, as it is often the preferred method to express domains, ranges, and intervals of increase or decrease for functions.